CONTROL CONFIGURATIONS IN BUYER-SUPPLIER RELATIONSHIPS: ENVIRONMENT -BUYER ORGANISATION -GOALS AND MODES OF CONTROL

Keywords:

Considering the growing externalisation of strategic activities, the problem of the control of buyersupplier relationships is crucial. Therefore, researchers usually propose modes of control that are adapted to various environments. However, the organisations are often considered as "black boxes" whose goals are unspecified. This paper examines buyer-supplier control configurations that take into account the organisation of buying firms and their goals toward their suppliers. This research is based on six case studies conducted in the manufacturing industry (60 interviews). The outcome of the research is a matrix which represents four configurations of buyer-supplier control, based on the global purchasing environment of the buying firm (in terms of reciprocal dependence between the buyer and its suppliers). For each configuration, a type of purchasing organisation (structure and intra-organisational control of purchasing agents) and a principal goal for the buying firm are proposed: the lord-buyer wants to exert its power, the partner-buyer aims at assuring goal congruence with its suppliers, the vassal-buyer tries to reduce uncertainty and the market-buyer seeks to grasp opportunities on the market. For each configuration, the modes of control that the buyer exerts on its suppliers -in terms of means, objects of control, influence strategies of the buyer (more or less coercive) and suppliers reactions-are coherent with the main goal of the buyer.

INTRODUCTION

Outsourcing is a form of inter-firm relationships that has become a crucial subject of study. In many industrial sectors, the economic performance of firms that outsource (here the buyer firms) is due, to a large extent, to the suppliers that contribute roughly 80% pf the added value of products sold. Moreover, firms are implicated in the long term with suppliers that conceive and realise their strategic components. As a result, buyer firms need to manage their relationships and more specifically to organise the control of their suppliers to reach their goals.

Research studies on inter-organisational control have proliferated during the last decade. In particular, considering the diversity of relationships, authors have developed typologies that are adapted to the characteristics of the transactions, environment and parties [START_REF] Dekker | Control of inter-organizational relationships : evidence on appropriation concerns and coordination requirements[END_REF][START_REF] Dumoulin | Marché, bureaucratie et clans dans les réseaux stratégiques[END_REF][START_REF] Hakansson | Accounting and network coordination[END_REF][START_REF] Van Der Meer-Kooistra | Management control of interfirm transactional relationships : the case of industrial renovation and maintenance[END_REF].

Though these works have added the growing knowledge of the design of control systems in outsourcing relationships, the authors generally consider the firms as "black boxes" whose goals are underspecified. This paper examines control in outsourcing relationships and takes into account the organisation of buyer firms and their goals toward their suppliers as well as the modes of control they exert on them. We want to give some theoretical and empirical evidence to answer to the question: Why and how buyers control their suppliers?

This research is based on six case studies conducted in the manufacturing industry. The purchasing environment of theses firms is characterised by the degree of reciprocal dependence between this firm and its suppliers. The internal purchasing organisation of thses firms has been studied (in terms of structure and intra-organisational control of purchasing agents) and the control they exert over their suppliers. This paper is structured in three parts. The first examines the three main views on interorganisational control in the literature (transactional, relational and power-dependence views) and the conceptual framework of the study. The second is dedicated to the presentation of the qualitative methodology. The third presents the four buyer-supplier configurations that have been observed: lordship, vassalage, market and partnership.

CONCEPTUAL BACKGROUND

Control in Inter-organisational Literature

Three main views structure inter-organisational control literature [START_REF] Heide | Interorganizational Governance in Marketing Channels[END_REF]: the Transactional view, the Relational view and the Power and Dependence view. In each of these views, theorists propose modes of control that aim to one ore several goals.

Goals and Modes of control in the Transactional View

The transactional view gathers theorists who analyse exchanges as economic transactions. In this view, the transaction cost theory (TCT) developed by [START_REF] Williamson | The Economic Institutions of Capitalism[END_REF] indicates that the characteristics of transactions (uncertainty, frequency, degree of specific assets i ) as well as the context in which they occur (small numbers, opportunism and bounded rationality) entail transaction costs. TCT proposes three governance structures and contracts to cap transaction costs: the market (classical contract), the hierarchy (personalised contract), or the hybrid form (neo-classical contract). The market relies on free competition to ensure control, the hierarchy relies on authority (fiat), and the hybrid form relies on long-term contracts. The control modes associated with these governance structures aim to gain efficiency, reduce transactional risk [START_REF] Dekker | Control of inter-organizational relationships : evidence on appropriation concerns and coordination requirements[END_REF] and protect against the opportunism of exchange partners. Also within transaction view, agency theory raises the problem of the management of opportunism in the context of uncertainty and diverging interests between the parties. [START_REF] Celly | Outcome-Based and Behavior-Based Coordination Efforts in Channel Relationships[END_REF] have tested the propositions of this theory in buyer-supplier relationships. They show that the more complex the product of the exchange is, the environment uncertain and the supplier difficult to replace, the more the buyer favours control of behaviours rather than control of results.

Goals and Modes of Control in the Relational Exchange View

Scholars of the relational exchange view develop a social conception of exchanges. Based on social exchange theory [START_REF] Blau | Exchange and Power in Social Life[END_REF][START_REF] Thibaut | The Social Psychology of Groups[END_REF], on Macaulay's seminal study about non-contractual relationships, and on Macneil's work about the distinction between discrete and relational exchange, theorists from the relational view study control mechanisms (generally social ones) that govern exchanges and better the efficacy of inter-firm relationships [START_REF] Larson | Network Dyads in Entrepreneurial Settings : A Study of Governance of Exchange Relationships[END_REF]. These mechanisms develop according to different phases [START_REF] Dwyer | Developing Buyer-Seller Relationships[END_REF]. In the beginning, cooperation only concerns risk-free operations and relies on classical contracts. Then, the success of these exchanges encourages the partners to engage in riskier operations and to deepen cooperation. Repeated interactions between individuals lead to the emergence of explicit rules (communication procedures, rules of information sharing) and implicit rules (honesty, openness, equity, reciprocity) that consolidate cooperation [START_REF] Larson | Network Dyads in Entrepreneurial Settings : A Study of Governance of Exchange Relationships[END_REF]. These rules are actually relational norms that structure exchanges and provide the required coordination. So, informal control that progressively replace formal devices [START_REF] Ring | Development Processs of Cooperative Interorganizational Relationships[END_REF] are above all a means to achieve common goals in the long term. Firms cooperate to achieve mutual gain rather than behaving opportunistically to satisfy their individual interest alone.

Goals and Modes of Control in the Power Dependence View

Considering that firms are dependent on the environment where they find the resources they need to achieve their goals, some scholars have applied the concept of dependence developed by [START_REF] Emerson | Power-Dependence Relations[END_REF] and [START_REF] Blau | Exchange and Power in Social Life[END_REF] in inter-personal context to inter-organisational exchanges. [START_REF] Pfeffer | The External Control of Organizations[END_REF] have developed the Resource Dependence Theory that proposes diverse strategies of adaptation and avoidance given the situations of uncertainty and dependence faced by the organisations. In particular, the authors suggest structuring exchanges by establishing formal or semi-formal inter-organisational links such as contracts, joint ventures, or mergers [START_REF] Ulrich | Perspectives in Organizations: Resource Dependence, Efficiency and Population[END_REF]. The response put forward by organisations to the problem of uncertainty and dependence is to stretch coordination with the appropriate partners and to create "negotiated environments" 1 . Here, the organisation's performance is determined by its effectiveness, whereas the transactional view emphasizes efficiency. In addition, the scarcity of resources does not encourage organisations to get to know each other better or to cooperate in a reciprocal manner, as the relational exchange view suggests, but to develop strategies of influence (or, on the contrary, strategies that limit influence) over the organisations that hold the resources (Oliver, 1990). Also within the power dependence view, researchers studied the way power may be exerted.

Following [START_REF] French | The Bases of Social Power[END_REF], Frazier and his colleagues study coercive influence strategies -threat, legal argument, promise-and non-coercive influence strategies -request, recommendation, exchange of information-that impact actions [START_REF] Frazier | Dealer Dependence Levels and Reciprocal Actions in a Channel of Distribution in a Developing Country[END_REF][START_REF] Frazier | Perceptions of Interfirm Power and Its Use within a Franchise Channel of Distribution[END_REF]. So, in this view, modes of control (e.g. contracts, coercive or non coercive strategies) aim to limit the influence of the organisations that hold the critical resources or, in the contrary, to exert power on dependent organisations.

Table 1 presents a summary of the three theoretical views presented.

1 Term borrowed from [START_REF] Cyert | A Behavioral Theory of the Firm[END_REF]. These different views add to the knowledge of inter-organisational control. We suggest integrating their contribution in a conceptual framework based on organisational control concepts.

Views

The application of organisational control concepts to inter-organisational control

Inter-organisational control conception

Following [START_REF] Flamholtz | Effective organizational control: a framework, applications and implications[END_REF]'s conception of organisational control, we define the control the buyer exerts on its supplier as the process by which the buyer influences the actions of the supplier to reach its goal. This conception of control entails four ideas. First, control is a process. Each episode of the relationship (for example each transaction) occurs within an historical and sociological context (contribution of the relational exchange view). Second, the interests of the parties can diverge or converge depending on the situation. Third, control is finalised toward the goals of the buyer. We postulate that the buyer knows its goal. Though it is not always able to give some precise objectives to its supplier, it can give it a direction.

Fourth, control can be exerted through formal and / or informal mechanisms. In interorganisational literature, the relation between formal and informal control is much disputed.

Scholars of the transactional view favour a formal conception of control to fight against the opportunism of the partner and to reduce uncertainty. On the contrary, theorists of the relational exchange view suggest that trust and relational norms give the required coordination of exchanges whereas formal control leads to suspicion [START_REF] Nooteboom | Effects of Trust and Governance on Relational Risk[END_REF]. Some others consider that trust and control are complementary to favour confidence (Das and Teng, 1998;[START_REF] Guibert | La complémentarité entre contrôle « formel » et « contrôle informel » : le cas de la relation client-fournisseur[END_REF]. We do not favour a priori one or the other conception but we will try to observe formal mechanisms as well as informal processes.

Since the conception of buyer-supplier control is clarified, we suggest to expose our conceptual framework.

Conceptual framework

The objective of this research is to study buyer-supplier control configurations that take into account the organisations of the buyers and their goals. We propose to structure our conceptual framework in three points: first, the purchasing environment of the buyer; second its organisation and third, the control process over the suppliers.

Purchasing environment of the buyer

The environment of buyer-supplier relationships is often differentiated according to the characteristics of the transactions, the characteristics of the parties or the degree of uncertainty [START_REF] Langfield-Smith | Management control systems and trust in outsourcing relationships[END_REF][START_REF] Van Der Meer-Kooistra | Management control of interfirm transactional relationships : the case of industrial renovation and maintenance[END_REF]. In the study, we have not opted for categories a priori to describe the purchasing environment of the buyer. We will rely on the empirical data to characterise it.

Organisation of the buyer

Two aspects of the organisation will be studied: the structure of the buyer and the intraorganisational control of the purchasing agents.

The structure can be defined as "a relatively stable allocation of tasks and roles that create a pattern of interrelated activities that allow the firm to conduct and coordinate its activities" (Desreumaux, 1992, p. 50). The structure explains the architecture of the control the buyer exerts over its supplier. In this research, the structural form will be analysed as well as the processes of decision making that can be more informal.

The intra-organisational control of the purchasing agents is defined as the process by which the buyer organisation influences the actions of its purchasing agents to reach its goals. The purchasing agents are key actors of the suppliers control process since their role is to ensure the interface with the suppliers [START_REF] Ring | Development Processs of Cooperative Interorganizational Relationships[END_REF]. This is why it seems crucial to understand how their firm controls them (objectives assigned, evaluation of their performance…).

The control of the buyer over its suppliers [START_REF] Chiapello | Les typologies des modes de contrôle et leurs facteurs de contingence : un essai d'organisation de la literature[END_REF] organised the literature on organisational modes of control along several dimensions: the objects of control, its processes, its moments, the means of control and the reactions of the controlled party. Each of theses dimensions respond to a question: What?

How? When?... However, the dimension "goals of control" (the question Why?) is excluded from the typology. This dimension is fundamental because it sheds light on the strategy of the one who exerts control and gives sense to the other dimensions. In the organisational literature, three main goals of control can be identified:

-Ensure goal congruence between the parties: control aims to ensure coherence and cohesion between the members of the organisation so that organisational goals are achieved. Scholars who have worked on systems of coherence and coordination [START_REF] Bouquin | Le contrôle de gestion[END_REF][START_REF] Hopwood | Accounting and Human Behavior[END_REF], on integration mechanisms [START_REF] Lawrence | Organization and Environment : Managing Differenciation and Integration[END_REF] or on cohesion in organisations (Pech Varguez, 2003) stand by this perspective.

In inter-organisational literature, authors in the relational exchange view stress as well on the importance of the coordination of activities.

-Exert one's power: the one who controls tries to obtain behaviours from others that they would not have had otherwise. With reference to [START_REF] Dahl | The Concept of Power[END_REF]'s concept of power, [START_REF] Tannenbaum | Control in Organizations: Individual Adjustment and Organizational Performance[END_REF] suggests that control has a symbolic significance since it implies the superiority of the one, the inferiority of the other, the criticism of the one, the submission of the other… In the inter-organisational literature, to exert one's power is one of the goals of those who stand on the power dependence view.

-Reduce uncertainty: control aims to ensure that no bad surprise will occur. [START_REF] Merchant | The Control Fonction of Management[END_REF] supports this perspective. Good control is the insurance that the realisations will match up with the plans. In inter-organisational literature, transaction cost theorists and resource dependence theorists propose modes of control that aim at attaining this goal.

These three main goals that are identified from organisational control literature can be compared to those of the three main inter-organisational views. Our organisational conception of control allows us to integrate the different views in our conceptual framework and to study modes of control that respond to one or several goals according to the empirical situations.

The modes of control will be analysed according to three dimensions proposed by [START_REF] Chiapello | Les typologies des modes de contrôle et leurs facteurs de contingence : un essai d'organisation de la literature[END_REF]: the objects of control (results, working processes, behaviours, strategic choices), means of control (market, organisation, individual interactions) and the reactions of the controlled party (here the supplier).

METHODOLOGY

This research draws from six case studies of French buyer firms. These case studies enabled us to consider complex control processes, to describe the organisation of buyer firms and their goals. Since these phenomena remain largely unexplored, it seemed judicious to carry out a qualitative empirical analysis [START_REF] Yin | Case study research: design and methods[END_REF].

Each of the case studies that has been chosen for this research evolves in a purchasing environment that can be described in terms of reciprocal dependence between the buyer firm and its suppliers. A preliminary study was conducted in an automobile equipment manufacturer. The results showed that four types of buyer-supplier relationships (partnership, market, submission of the buyer, domination of the buyer) could be distinguished according to the reciprocal dependence between the firms. We contacted 13 firms in diverse industrial sectors and conducted exploratory interviews with people from each firm. We observed that 10 of theses firms developed relationships with their suppliers that could be mainly classified in one of the four types. According to this type, the organisation of the buyer and the control that was exerted over the suppliers seemed different. Among the ten potential cases, three firms did not wish to take part in further interviews given the time required by an in-depth study. Another firm experienced serious financial difficulties (liquidation), cutting the study short. We therefore possess detailed data on six cases from diverse industrial sectors:

automotive (Autoplus and Consauto), aeronautics (Airindustry), pharmaceutical (Pharma), agrifood (Agri) and telecommunication (Telecom).

For each of the cases, we carried out interviews with the purchasing directors and purchasing agents. We questioned them about the environment of buyer-supplier relationships in which their firm evolved. We then asked them to choose the supplier they considered to be the most representative of their portfolio and asked them to comment on the extent of the asymmetry of their dependence and that of the chosen supplier2 . We questioned each purchasing agent on his expectations with regard to the selected supplier, assigned goals, modes of monitoring, modes of assessment, relational modes, as well as on the reactions and strategies of the supplier. We wanted to focus the interview on one particular relationship so as to avoid commentary on the management control of suppliers in general. In parallel, we interviewed suppliers about the relational modes fostered with the buyer and about the strategies that they implement with regard to the buyer. 60 people were interviewed.

Finally, we consulted and studied internal corporate documents (e.g. supplier follow-up spreadsheet, supplier assessment grids, framework contracts, etc.) and external firm documents (press reviews, financial statements, etc.). All the data was analyzed according to the method recommended by [START_REF] Miles | Analyse de données qualitatives, recueil de nouvelles méthodes[END_REF]. We constructed relevant categories from our analysis of the literature and completed them using field data. The data were coded using the software NVivo, which enabled us to lay down intra-site, then inter-site, matrices in order to draw out patterns of analysis.

CONTROL CONFIGURATIONS IN BUYER-SUPPLIER RELATIONSHIPS

The analysis of the data led us to create a matrix (figure 1) on which cases can be positioned according to the two axes that were identified (dependence of the buyer / dependence of the supplier).

The dependence of the buyer was evaluated according to:

-the concentration of the potential suppliers -the concentration of the exchanges with the suppliers -technological, industrial and regulations constraints

The dependence of the suppliers was evaluated according to:

-the concentration of the buyers -the concentration of the exchange with the buyer

Figure 1: Position of the cases on the matrix

Four control configurations in buyer-supplier relationships were identified. For each configuration, we observed coherence between the purchasing environment of the buyer firm, its purchasing organisation (structure and intra-organisational control of the purchasing agents) and control that the buyer firm exerts on its suppliers (principal goal and modes of control).

Low interdependence between the buyer and its suppliers: the market configuration

Figure 2 presents the articulation between the purchasing environment of the buyer firm, its organisation and the control it exerts over its suppliers in the market configuration.

Figure 2: Market configuration

Purchasing organisation

The structure A buyer is, overall, in a situation of low dependence with its suppliers because it buys standard products that many potentially substitutable suppliers can produce. It has interest to centralise its purchase to take advantage of the volume and to diminish its cost of structure [START_REF] Gadde | Professional Purchasing[END_REF]. The purchasing structure of Telecom (which is in a market configuration), is actually centralised to the level that is considered to be relevant (the zone or the group).

Intra-organisational control of purchasing agents

The purchasing agents of Telecom are evaluated on their results, principally in terms of economic savings. Their objectives are clear, formalised, and measurable by the information system. The objectives are the same for every purchasing agent, but the amount of the economic gains depends on the market anticipations concerning the different purchase domains. Another objective assigned to the purchasing agents concerns the logistical performance of the suppliers due to uncertainty on the volumes and industrial constraints of the production units. Purchasing agents have principally a role of rationalisation which consists of reducing costs and rationalising the flows [START_REF] Gadde | Professional Purchasing[END_REF].

M a ke th e su p pliers in co m p etitio n C o n vin ce the oth er in terna l a cto rs to sim p lify th eir d em a n d s a n d to ch a n g e su p p liers wh en req uired

To obtain these results, the purchasing agents have to convince the prescriptors (e.g. the engineers of the research units) and the users (e.g. the personnel of the production units) who are their internal client that their choices are relevant. As a result, the purchasing director requires the purchasing agents to adopt certain behaviours (communication, listening of the needs) toward the other internal actors so that they integrate into their global purchasing policy some local preoccupations.

Control the suppliers

Taken into account the low interdependence relationship between the buyer and its suppliers, the buyer Telecom tries to grasp any opportunity on the market3 at any time. The characteristics of market control have already been described in the literature (Langfield-Smith and Smith, Van der Meer-Kooistra and Vosselman, 2000): periodical competitive biddings, no detailed contracting, payment based on standardised activities or output.

However, our study suggests that organising regular competitive biddings does not mean that the buyer will change from one supplier to another. Threats of production transfer are certainly frequent during the negotiations, mainly for economic reasons, but the putting into practise of these threats seems to be rare. Even when the transfer is possible, the other actors in the buyer firm (prescriptors, users) prefer continuity to rupture of the relationship and they resist the change of suppliers.

Low dependence of the buyer, high dependence of the suppliers: the lordship configuration

Figure 3 presents the articulation between the purchasing environment of the buyer firm, its organisation and the control it exerts over its suppliers in the lordship configuration.

Figure 3: Lordship configuration

Purchasing organisation

The structure

The purchasing structure of Agri (which stands in the lordship configuration) is, as Telecom's, centralised to the "relevant" level (the geographical zone or the group) so that it can take advantage of the volume effect and diminish its cost structure.

Intra-organisational control of purchasing agents

The purchasing agents of Agri are evaluated on their results, principally in terms of economic savings. Their objectives are clear, formalised, and measurable by the information system. However, the purchasing agents cannot achieve these results alone. They need that prescriptors (marketing, R&D) to simplify their product and that the users (production units) agree on changing their suppliers if necessary. As a result, the purchasing agents are also evaluated on their ability to communicate within the group, to convince the others of the relevancy and to create an internal network.

Control over the suppliers

In light of its favourable position of asymmetrical dependence, Agri seeks to exert its power over its suppliers to get the most out of exchange, in a way similar to a lord exercising his rights over his vassals. This behaviour is consistent with the analyses of Dwyer and Walker (1981), [START_REF] Kale | Dealer Perceptions of Manufacturer Power and Influence Strategies in a Developing Country[END_REF], and [START_REF] Provan | Interorganizational dependence and control as predictors of opportunism in dealer-supplier relations[END_REF] within the power and dependence view and coincides with the postulates of the transactional view: he who holds the power does not hesitate to exploit the relationship and show himself to be opportunistic.

The exercise of power implies a formal control on the results (price, quality, on-time delivery), on working processes, on behaviours, and on strategic choices. Two modes of control are developed in tandem: recourse to the market and organisational control. On the one hand, market is a means to ensure the lowest price (e.g. inverse auctions on the Internet).

On the other hand, organisational devices are also developed. Agri regularly redefines goals in terms of price, quality, delivery conditions, and volumes. It also expects suppliers to develop a strategy (in terms of innovation, of production organisation, etc.) in step with its demands and imposes compliance with relational norms in its favour (providing services, being proactive in seeking new sources of productivity, showing flexibility, being totally transparent on the breakdown of costs, etc.). In parallel to the classic process of control (objectives, monitoring, assessment/sanction), Agri is developing a strategy of encroaching its suppliers. It is ensuring that it is treated as a favoured buyer by performing follow-up audits and spotchecks at supplier sites. These devices aim to verify in situ the truthfulness of information provided by commercial counterparts (notably information regarding the breakdown of supplier costs, the market shares of different buyers, etc.) and to monitor competition. Agri's encroachment is limited only by its cost (mainly in terms of human resources) compared to the benefits that it may draw from the process.

Agri's suppliers, having few action levers over their buyer, are also developing active strategies to encroach on production units (by being present with operators, workshop supervisors, and directors) in order to develop a network of relationships and favour Agri's relational competencies. They apply themselves to responding as best they can to the concerns of production units while trying to glean information on potential competitors and on new projects for which they could position themselves. Moreover, they emphasize their relational behaviour to foster the buyer's benevolence; they communicate intensively and try to develop good personal relationships with the different Agri counterparts. This supplier strategy of influence is in step with research findings stemming from the relational exchange view: the more individuals communicate, the more trust and relational norms develop, which, in fine, fosters cooperation in the long term and discourages the buyer from threatening its supplier or switching production to other suppliers (Boyle et al., 1992;Joshi and Arnold, 1997;[START_REF] Morgan | The Commitment-Trust Theory of Relationship Marketing[END_REF]. However, our observations go further, suggesting that the affective influence of suppliers results from pure calculation on their part. Where the authors of the relational exchange view do not discuss the authentic or calculated character of relational behaviours in cooperation, and in particular of goodwill trust, we observe, on the contrary, that relational strategies stem from a marked intention that goes far beyond goodwill trust.

High interdependence between the buyer and its suppliers: the parnership configuration

Figure 4 presents the articulation between the purchasing environment of the buyer firm, its organisation and the control it exerts over its suppliers in the partnership configuration.

Purchasing organisation

The structure A buyer is, overall, dependent on its supplier because its demands are to specific to have many potential suppliers (due to the technology of the products, of the processes, due to the required production capacity or international implantation, etc.). A decentralised structure is coherent with this complex type of purchase that is usually specific to a project (or to a business unit). In Consauto and Autoplus (which stand in this partnership configuration), the purchasing structure is effectively decentralised: local purchasing agents are very close to the research units for each project cell. However, to exploit similarities between the projects and to take advantage of the volume effect, the purchasing structure is also centralised. Some global purchasing agents are in charge of suppliers in their domain for the whole firm. This double structural characteristic (centralisation / decentralisation) implies in both cases a O r g a nisa tion -m a trix str u ctur e w ith su pp or t fu n ction s (ex p er ts) -c on tr ol of p u r ch a sin g a g en ts fo cu sed on th eir r esu lts (m a in ly e c on om ic g ain s) an d on th eir beh a viou r s tow a r d th eir su p p lier s

P a rtn e rsh ip co n fig u ra tio n T a k e ad v a nta g e of th e v olu m e e ffe ct A llo c ate re so u rc e s to m o nito r a n d e v a lua te th e sup p lie rs S e le ct a b le su pp lie rs Im p lic ate in th e lo ng te rm C o o rd ina te in te rd e p e nd e n t a ctiv itie s E n su re c o h e sio n b e twe e n p roje ct te a m s R e d u ce co sts P ilo t su p plie rs d u rin g d e v elo p m e nt p h a se D e v e lo p c oo p e ratio n

matrix structure. Moreover, to better control their suppliers on different aspects (quality process, cost reductions…), Consauto and Autoplus reinforced this matrix structure with some experts (or support functions) from the purchasing department or those that are detached from other departments to audit the suppliers.

Intra-organisational control of purchasing agents

In both cases, local purchasing agents are responsible for the operational control of the suppliers. They are asked to implement the monitoring of the risks in the project phase and the monitoring of the results (on-time delivery, quality results, and price reductions) in the production phase. Global purchasing agents are required to obtain price and cost reductions and to ensure the quality of the However, the purchasing agents who are evaluated on the economic results (cost reductions) could be tempted to obtain great price reductions in the short term by favouring very coercive strategies (threatens, sanctions) over dependent suppliers, without taking into account the dependence of the buyer in the long term. To prevent from this risk, the purchasing agents are required by internal directives (in Consauto and Autoplus) to behave positively (to be honest, ethical, to respect their obligations) to limit/reduce risk of their suppliers. Additionally, close attention seems to be paid to any negative information that come from suppliers (complaints, discontents, …).

Control over the suppliers

In a partnership configuration, the buyer aims to ensure goal congruence with its suppliers. This goal includes the search for coherence as well as the willingness to create a certain degree of cohesion between the teams.

In the selection phase, the buyer exerts an enlarged bureaucratic control over its suppliers on its capacity, on its performance about quality, logistics, industrialisation, international implantation, innovation, strategic choices. The experiences of the relationships and the reputation of the suppliers are taken into account. Selection mechanisms (e.g. numerous audits, evaluation of strategic choices, of the financial health) allow the buyer to better know its suppliers, to be sure that it has appropriate attitudes (in terms of quality approach, production capacity, innovation capacity, etc.) which is, according to [START_REF] Dekker | Control of inter-organizational relationships : evidence on appropriation concerns and coordination requirements[END_REF] a means to reduce problems of control and to instill confidence. Selection mechanisms appear in fact to favour ex ante coherence between the firms. However, the selection of the suppliers is not a economico-rational process. The choice of the suppliers are interlinked in a political game between the actors (e.g. between the purchasing agents, the engineers, the project directors) who seem to favour one criterion or another to influence the decision.

During the cooperation, the devices that are designed by the buyer to control its partner are numerous and complex. They are implemented by the different actors of the buyer firm (local, global purchasing agents, experts…). Theses devices are certainly costly for the buyer firm (mainly in term of human resources) but they may generate direct savings (for example productivity savings gained from the various audits) and indirect benefits (better informal knowledge of the supplier). The implementation of theses devices gives the different actors of the buyer firms the opportunity to glean information on the strategy, on the real capacity of the supplier, etc. Moreover, we observed between the project teams relationships relational norms (mainly solidarity, information sharing) and trust-based relationships, which does not exclude conflicts. This confidence atmosphere that stems from the complementarity between formal and informal control seems to favour in fine the inter-organisational cohesion.

High dependence of the buyer, low dependence of the suppliers: the vassalage configuration

Figure 5 presents the articulation between the purchasing environment of the buyer firm, its organisation and the control it exerts over its suppliers.

The structure

As seen before, a decentralised structure is consistent with complex types of purchase (due to the technology of the production, of the processes, etc.), which implies for the buyer to be

V assa lage con figu ration

A llocate resources to m onitor suppliers E nsure the continuity of the suppliers A nswer to the dem ands of the prescriptors and of the users C ontractualise with suppliers M onitor the supplying of the production units dependent on its supplier. This analysis is confirmed by the cases Airindustry and Pharma.

However, the parallel centralisation that leads to a matrix structure has not been observed because global synergies between the different business units are considered to be too limited.

Intra-organisational control of purchasing agents

In the vassalage configuration, purchasing agents' performance is not assessed by the economic savings. These are, anyway, limited since dominant suppliers do not accept decreases in their prices. The purchasing directors' main request from their subordinates is to achieve operational results:

-to ensure the security (in terms of quality and regularity) of the supply of the production units; -to develop purchasing methods (to design and implement indicators and devices to monitor the performance of the suppliers, to develop internal reporting, to create "decision making aids" ) -to get involved upstream during the development stage phase at the engineers' side.

Control over the suppliers

In light of their unfavourable position of asymmetrical dependence, vassal buyers seek to reduce their uncertainty concerning the quality and the continuity of the supplying by trying to structure their exchanges through contracts. These results are consistent with the propositions of the dependence resource theory [START_REF] Pfeffer | The External Control of Organizations[END_REF] and those of the transaction cost theory [START_REF] Williamson | The Economic Institutions of Capitalism[END_REF] in situations of high dependency. The suppliers, however, only accept the contracts required by regulation. They reject any mode of incentive, monitoring or sanctioning through contract (no late-delivery penalties or commitments on price...). Airindustry and Pharma have also failed to develop behavioural control (for instance, through auditing processes) as recommended by agency theory whenever the product is complex and uncertainty high. Suppliers refuse all on-site spot checks and other forms of observation that they consider to be intrusive. The only controllable and acceptable elements for suppliers remain the respect of quality and delivery deadlines since these are considered "health factors" of cooperation. To compensate for the lack of formal supplier control tools, Airindustry and Pharma rely on more informal mechanisms. They are seeking to develop good, sometimes even friendly, personal relationships with their counterparts by multiplying the number of contacts, making themselves available, and presenting the prospect of future cooperation. By doing so, they create the conditions of "affective trust" (Jeffries and Reed, 2000;[START_REF] Mcallister | Affectand cognition-based trust as foundations for interpersonal cooperation in organizations[END_REF]. Thanks to this relational strategy, buyers are attempting to favour suppliers' commitment, ensure a certain flexibility and continuity of supply, as well as obtain exclusive information on new products, competitors, and even sometimes cost structures.

CONCLUSION

In this research, we have attempted to study the control that buyers exert over their suppliers according to the characteristics of the purchasing environment of the buyer. Four configurations have been identified. For each one, the buyer organisation (in terms of structure and intra-organisational control of purchasing agents) and the control over the suppliers are consistent with the purchasing environment of the buyer firm. The Table 2 presents a synthesis of the results.

Internal control over the purchasing agents

On their results

Economic savings Economic savings

Economics savings

Operational results (quality, on-time delivery)

On their behaviours

Internal Internal Toward the suppliers -Internal -Purchasing methods

Control over the suppliers (principal goal for the buyer)

To grasp the opportunities on the market

To exert its power

To ensure goal congruence

To reduce uncertainty

Table 2: buyer-supplier control configurations

For each configuration, a principal goal for the buyer firm has been identified and gives sense to the modes of control that are designed and implemented: the lord buyer aims to exert its power; the partner buyer seeks to ensure goal congruence with its supplier; the vassal buyer tries to reduce its uncertainty; and the market buyer aims to grasp any opportunity on the market. This latter goal did not appear in the initial typology of control goals. It is particularly adapted to the outsourcing context since it implies market control. It may also be relevant in an organisational context. Some devices already aim at attaining it (for example benchmarks, the SWOT devices, the balanced scorecard) although the theorists in organisational control do not explicitly express it. In this perspective, the control system can be considered as an alert system which is turned to outside (i.e. the environment) and not to inside (i.e. the internal organisation) so that managers be able to grasp the best opportunity of their environment.

Moreover, by studying the organisation of the buyer firms, we observed that many internal actors are concerned with buyer-supplier relationships. The interests of these actors are not necessarily converging: purchasing agents are often interested in prices (according to their objectives), prescriptors (development engineers) seem to lay stress on technology and personnel in production units prefer to ensure the quality of the supply. Thus, a transfer of production that would be economically relevant can be hindered because the engineers or the personnel of the production units refuse it. As a result, the strategy of the suppliers is to cosy up the different department of the buyer firm. They create the conditions of an affective influence on these actors. To better understand this form of supplier control over buyers and the complexity of the internal coordination in buyer firms, it could be interesting to deepen the study of buyer organisations. Similar research could also be conducted on supplier firms.

Finally, we based our methodology on case studies in order to analyze a phenomenon that remains mostly unexplored. This choice entails the limitation of having findings that are singular and cannot be generalized. We envisage putting these initial findings (or a part of these findings) to the test on a broader population.

BIBLIOGRAPHY

P

  urch a sin g en viron m en t L o w in terd ep en d en ce (m a n y p o ten tial b u yers a nd su p pliers) C on tr ol over sup p liers G oal: T o grasp op p ortun ities on th e m ark et O rg a nisa tion -cen tralised stru cture -con trol of p u rch asin g ag en ts fo cu sed on th eir resu lts (econ om ic g ain s) an d on th eir in tern al beh aviou rs M ark et C on figu ration T a ke ad va nta g e of th e volu m e effect O b ta in th e b est p rices D im in ish th e co st of th e p u rcha sin g stru ctu re C h o o se the m o st co m petitive su p p liers o n th e m a rket M a ke th e su p pliers in co m p etitio n C o n vin ce the oth er in terna l a cto rs to sim p lify th eir d em a n d s a n d to ch a n g e su p p liers wh en req uired P urch a sin g en viron m en t L o w in terd ep en d en ce (m a n y p o ten tial b u yers a nd su p pliers) C on tr ol over sup p liers G oal: T o grasp op p ortun ities on th e m ark et O rg a nisa tion -cen tralised stru cture -con trol of p u rch asin g ag en ts fo cu sed on th eir resu lts (econ om ic g ain s) an d on th eir in tern al beh aviou rs M ark et C on figu ration T a ke ad va nta g e of th e volu m e effect O b ta in th e b est p rices D im in ish th e co st of th e p u rcha sin g stru ctu re C h o o se the m o st co m petitive su p p liers o n th e m a rket

  sh ip con figu ration O btain the best conditions from the suppliers (price, service, innovation, etc.) Take advantage of the volum e effect O btain the best prices D im inish the cost of the purchasing structure O rganisation -cen tralised structure -con trol of purchasin g agen ts focused on th eir results (econ om ic gain s) and on th eir internal beh aviours M ake the suppliers in com petition C onvince the other internal actors to sim plify their dem ands and to change suppliers when required C ontr ol over suppliers G oal: T o exert it's pow er Purchasing environm ent H igh unilateral depen den ce of th e suppliers (m any potential suppliers, few potential buyers) L o rd sh ip con figu ration O btain the best conditions from the suppliers (price, service, innovation, etc.) Take advantage of the volum e effect O btain the best prices D im inish the cost of the purchasing structure O rganisation -cen tralised structure -con trol of purchasin g agen ts focused on th eir results (econ om ic gain s) and on th eir internal beh aviours M ake the suppliers in com petition C onvince the other internal actors to sim plify their dem ands and to change suppliers when required C ontr ol over suppliers G oal: T o exert it's pow er

Figure 4 :

 4 Figure 4: Partnership configuration

P

  ur c h a sin g e n vir on m e n t H ig h in ter d ep en d en ce (fe w p o ten tial b u y ers a nd su p pliers, b ila teral sp ec ific inv estm en ts) C on tr ol ove r sup p lie r s G oa l: T o en su r e g oa l c on gr u en ce

Figure

  Figure 5: Vassalage configuration

P

  urchasing environm ent H igh unilateral depen den ce of th e bu yer (few potential suppliers, m any potential buyers, very specific products C ontr ol over suppliers G oal: T o reduce un certainty O rganisation -D ecen tralised structure -C ontrol of purchasin g agen ts focused on operation al results (techn ical aspects are dealt by specialists), on th eir internal beh aviours and on th e developm en t of purch asing m eth ods

Table 1: Inter-organisational Control according to the Principal Theoretical Views

		Transactional	Relational	Power Dependence
	Unit of analysis	Transaction	Relation	Organisation
	Objectives of the parties	Seek their personal	Motivation to	Exposure to the
		economic interest	cooperate and share the	demands of those who
			benefits of cooperation	hold resources and to
			fairly	their coercive influence
	Postulates on the goal	Interest conflicts	Goal congruence in the	Interest conflicts
	congruence between the	between the parties	long term between the	between the parties
	parties		parties
	Performance criteria	Efficiency	Developing	Effectiveness
			cooperation and its
			benefits in the long
			term shared out
			according to equity
			norms
	Control goals	Reducing transactional	Strengthening	Diminishing
		risks; protecting	cooperation over time;	dependence on
		against opportunism	coordinating joint	resources; impacting
			actions	the partner's actions
	Control modes	Different types of	Trust and relational	Formal links
		contract associated	norms (support	structuring exchanges;
		with governance	provided, flexibility,	more or less coercive
		structures (market,	solidarity, information	strategies of influence
		hierarchy, hybrid	exchange)
		form); offsetting
		investments; result or
		behaviour controls

We adopted the scales validated byAndaleeb (1995),[START_REF] Heide | Interorganizational Governance in Marketing Channels[END_REF] and[START_REF] Lusch | Interdependency, Contracting and Relational Behavior in Marketing Channels[END_REF]. The buyer evaluated three items from 1 to 5 (from "do not agree at all" to "completely agree"). We added up the scores to obtain an overall score for each variable. We then re-dimensioned this score on a scale of 0 to 20 using the following calculation: (overall score -3)*20/12.

This goal did not appear in the goals of control identified from organisational control works. It stemmed back to the empirical research.


Peter Bettess

Joseph Shirron

Omar Laghrouche

Bernard Peseux

Rie Sugimoto

Jon Trevelyan

A numerical integration scheme for special ÿnite elements for the Helmholtz equation

Keywords: short waves, ÿnite elements, special ÿnite elements, semi-analytical integration, numerical integration, partition of unity method (PUM)

The theory for integrating the element matrices for rectangular, triangular and quadrilateral ÿnite elements for the solution of the Helmholtz equation for very short waves is presented. A numerical integration scheme is developed. Samples of Maple and Fortran code for the evaluation of integration absciss and weights are made available. The results are compared with those obtained using large numbers of Gauss-Legendre integration points for a range of testing wave problems. The results demonstrate that the method gives correct results, which gives conÿdence in the procedures, and show that large savings in computation time can be achieved.

INTRODUCTION

This paper is concerned with recent developments in solving the Helmholtz equation for short wave problems. It is written here for reference as

@ 2 @x 2 + @ 2 @y 2 + k 2 =0; ∇ 2 + k 2 =0
(1)

k is the wavenumber, given by k =2 = , where is the wave length. is the ÿeld variable of interest, which could be wave elevation, pressure, or an electro-magnetic potential, among many other possibilities.

Recently the method of Melenk and BabuÄ ska [START_REF] Babuä Ska | The partition of unity method[END_REF][START_REF] Melenk | The partition of unity ÿnite element method. Basic theory and applications[END_REF], for solving the Helmholtz equation has been applied by Laghrouche and Bettess to a range of di raction problems [START_REF] Laghrouche | Solving short wave problems using the partition of unity ÿnite element method[END_REF][START_REF] Laghrouche | Plane waves solution for the 2D-Helmholtz short wave equation[END_REF][START_REF] Laghrouche | Solving short wave problems using special ÿnite elements[END_REF][START_REF] Laghrouche | Short wave modelling using special ÿnite elements-Towards an adaptive approach[END_REF][START_REF] Laghrouche | Short wave modelling using special ÿnite elements[END_REF][START_REF] Laghrouche | Short wave modelling using special ÿnite elements[END_REF][START_REF] Laghrouche | Analysis of the conditioning number of the plane wave approximation for the Helmholtz equation[END_REF]. Their results demonstrate that the method can solve short wave di raction problems with a greatly reduced number of active variables. Compared with conventional ÿne ÿnite element meshes, the number of variables has been reduced by up to 96% in some cases. Other authors have experimented with similar, though not identical concepts. Plane waves were used in the approximation of integral equations in electromagnetic scattering by de La Bourdonnaye [START_REF] De | High frequency approximation of integral equations modeling scattering phenomena[END_REF][START_REF] De | Une mà ethode de discrà etisation microlocale et son application  a un probl eme de di raction[END_REF] under the name of Microlocal Discretization. Mayer and Mandel [START_REF] Mayer | The ÿnite ray element method for the Helmholtz equation of scattering: ÿrst numerical experiments[END_REF] presented a similar method with the name Finite Ray Element Method. They investigated the asymptotic convergence of the method when the number of the approximating plane waves is increased. Recently, Farhat et al. [START_REF] Farhat | A discontinuous Galerkin-PW Method for the solution of high frequency acoustic scattering problems[END_REF][START_REF] Farhat | A discontinuous ÿnite element method for the Helmholtz equation[END_REF][START_REF] Farhat | The discontinuous enrichment method[END_REF] have proposed The Discontinuous Enrichment Method in which the standard ÿnite element polynomial ÿeld is enriched by plane waves in the case of acoustic problems. Ortiz and Sanchez [START_REF] Ortiz | An improved partition of unity ÿnite element model for di raction problems[END_REF] have developed a three node wave ÿnite element based on the partition of unity model and have produced good results for cylinder di raction problems. Their integrals are evaluated using an analytically exact procedure, in which the elements are represented in a right handed cartesian co-ordinate system in which the local x-axis is the average of each pair of wave directions. Most authors reported ill-conditioning problems.

The motivation for the use of these new elements is to escape the limitations of conventional ÿnite element procedures for the Helmholtz equation. These are limited in respect of numbers of nodal points needed to discretize a wavelength and in terms of pollution error, [START_REF] Deraemaeker | Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions[END_REF][START_REF] Ihlenburg | Finite element solution to the Helmholtz equation with high wavenumber part I: the h-version of the FEM[END_REF][START_REF] Ihlenburg | Finite element solution to the Helmholtz equation with high wavenumber part II: the h-p-version of the FEM[END_REF]. At each ÿnite element node the potential is expanded in a discrete series of plane waves, each propagating at a speciÿed angle. These angles can be uniformly distributed or may be carefully chosen. They can also be the same for all nodes of the studied mesh or may vary from one node to another. It has been shown that such systems of plane waves form complete sets of functions for the Helmholtz equation [START_REF] Herrera | Connectivity as an alternative to boundary integral equations[END_REF][START_REF] Sã Anchez-Sesma | A boundary method for elastic wave di raction: application to scattering of SH waves by surface irregularities[END_REF].

An outline of the present method is now given. More information is given in the papers cited above. We start by introducing the standard linear ÿnite-element 'hat' functions N i satisfying N i (x j )= ij , for each vertex x j in the mesh. The set {N i } M 1 forms a partition of unity. We use this partition of unity to construct our approximation space by letting the hat functions 'window' a set of global functions j (x)=e ikx• êj , which are plane wave solutions of the Helmholtz equation propagating in the direction of the unit vector êj . (The global functions could just as well be chosen as j (x)=J 0 (k|x -x j |), where {x j } is a set of points not necessarily coinciding with the mesh vertices. They could also be chosen as j (x)=J 0 (k|x-x j |) P n (x • x j =|x||x j |). These functions also satisfy the Helmholtz equation. At any rate, the success of the partition of unity method, (PUM) is not predicated on the chosen functions being solutions of the Helmholtz equation-they merely have to be good at locally approximating the solutions.) If at each mesh vertex we take N uniformly spaced directions of propagation { êj } N 1 , then we seek to approximate the solution to Equation (1) as

(x) ≈ M i=1 N j=1 ij N i (x) j (x) (2)
A weak form of the problem is obtained in the usual way by multiplying Equation (1) by a function , integrating that product over the domain . We assume the domain to be the ÿnite annulus bounded by the closed curves 1 and 2 which would normally be the scattering surface and the outer boundary of the ÿnite element model, respectively, though this is not strictly relevant to the discussion. We then apply the divergence theorem to produce

(∇ • ∇ -k 2 ) d = 2 ∇ • n 2 d 2 - 1 ∇ • n 1 d 1 (3)
where n i is the outward-directed unit normal vector on i . We obtain a Galerkin ÿnite-element method or more precisely, a Galerkin PUFEM, by taking

(x)= M m=1 N n=1 mn N m (x) n (x) (4)
Our interest is not in the boundary integrals on the right-hand side, but in the integral over the domain . On substituting in the expansions for and , we see that a typical element of the coe cient matrix has the form

[∇(N m n ) • ∇(N i j ) -k 2 N m n N i j ] d (5)
Next we insert the identity ∇ j =ik êj j into the above equation to obtain the form

n j [(∇N m + ik ên N m ) • (∇N i + ik êj N i ) -k 2 N m N i ] d (6)
The hat functions are non-zero over only a few neighbouring ÿnite elements. We can further restrict our attention to the form of the integral over a typical ÿnite element E t ,

I = E t e ik( ên + ê j )•x [(∇N m + ik ên N m ) • (∇N i + ik êj N i ) -k 2 N m N i ] dE t (7)
The integral of Equation ( 7) is highly oscillatory because of the presence of the exponential terms of imaginary argument, which are trigonometrical functions. Since k may be large, the exponential function in Equation ( 7) may contain many wavelengths. Previous workers, when evaluating the above integral have resorted to very high order Gauss-Legendre integration.

In the examples shown by Laghrouche and Bettess [START_REF] Laghrouche | Short wave modelling using special ÿnite elements[END_REF] up to 120 by 120 Gauss-Legendre numerical integration is used. So the reduction in the number of active variables, in the system matrix, comes at the price of some computationally intensive numerical integrations over the domain of the elements. The time for these integrations may be comparable with the execution time for the system matrix equation solution. The Gauss-Legendre method assumes that the function to be integrated is of a polynomial form. But it is well known that polynomial representations of trigonometrical functions are not accurate and are also expensive to compute [START_REF] Hamming | Numerical Methods for Scientists and Engineers[END_REF]. Indeed it seems completely illogical to develop new types of ÿnite elements specially for short waves, which include a knowledge of wave behaviour, and then to ignore this knowledge, and revert to polynomials, at the integration stage. The challenge then is to develop better integration methods, which incorporate our knowledge of the plane wave functions. This should drastically reduce the time taken to form the element matrices. The following analysis demonstrates how the integrations may be carried out more e ciently, by developing semi-analytical integration rules.

At this point, we introduce the assumption that the local-to-global geometric mapping is a bilinear function of the element co-ordinates, which implies that the edges of the elements are all straight. ‡ For a quadrilateral element, the global cartesian co-ordinates x =(x 1 ; x 2 ) are related to the local element co-ordinates ( ; Á) on E t by

x( ; Á)=a t + b t + c t Á + d t Á (8)
where the coe cients a t , b t and c t are vectors of dimension 2. The Á term complicates the evaluation of the integrals. Full details for the case where d t = 0 are given in Reference [START_REF] Zienkiewicz | The Finite Element Method[END_REF].

Here, we are mainly concerned with triangular or rectangular ÿnite elements, for which d t =0.

In this case, we see that the expression in the square brackets in [START_REF] Laghrouche | Short wave modelling using special ÿnite elements[END_REF] above is a quadratic polynomial in ( ; Á). (This is because the hat functions are linear and the Jacobian of the geometric mapping is constant.) Consequently, the integral in [START_REF] Laghrouche | Short wave modelling using special ÿnite elements[END_REF] has the form

I = A t  2 p; q=0 f t pq Ê e ik( ên + ê j )•(a t +b t +c t Á) L ' ( ; Á) d dÁ (9)
where

(∇N m + ik ên N m ) • (∇N i + ik êj N i ) -k 2 N m N i = 2 p; q=0 f t pq L ' ( ; Á) (10)
A t is the area of ÿnite element E t , and  is the area of Ê, the standard rectangular or triangular element. L ' ( ; Á) is an interpolating polynomial, which takes unit value at an integration point, and is zero at all the others. For rectangular and quadrilateral elements it can be constructed from the product of Lagrange polynomials (see Equation [START_REF] Herrera | Connectivity as an alternative to boundary integral equations[END_REF]). In those cases L ' ( ; Á)=L p ( ) × L q (Á). If the same mapping were retained, but higher-order interpolation were adopted, the upper limits on p and q would, of course, increase. Finally, we introduce t =k( ên + êj ) • a t ; ÿ t =k( ên + êj ) • b t ; t =k( ên + êj ) • c t [START_REF] De | Une mà ethode de discrà etisation microlocale et son application  a un probl eme de di raction[END_REF] and substitute these into (9) to obtain

I = A t  e i t 2 p; q=0 f t pq Ê e iÿ t e i t Á L ' ( ; Á) d dÁ (12)
The analysis has now arrived at an integration rule of the following form:

I = A t  e i t 2 p; q=0 f t pq w pq (ÿ t ; t ) (13)
where w pq (ÿ; )= Ê e iÿ e i Á L ' ( ; Á) d dÁ (14) ‡ Higher-order mappings could be used, but the integrations would not be exact, which is the case in normal ÿnite elements.

For the rectangle, the evaluation of the integration weights is simpliÿed because the integral of products becomes a product of integrals,

w pq (ÿ; )= +1 -1 e iÿ L p ( ) d +1 -1 e i Á L q (Á) dÁ (15)
Both integrals on the right-hand side are easily evaluated by repeated application of integration by parts. The evaluation of the integration weights for the standard triangle is more complicated. In this case

w pq (ÿ; )= 1 0 1-Á 0 e iÿ e i Á L ' ( ; Á) d dÁ (16)
The inner integral can be evaluated through integration by parts. The outer integral, while complicated, can also be evaluated analytically.

RECTANGULAR FINITE ELEMENTS

In this case the relation between the local, ; Á and global (x 1 ; x 2 ) co-ordinates given in Equation ( 8) simply becomes

x 1 = a 2 (1 -) x 2 = b 2 (1 -Á) (17)
where a and b are the dimensions of the rectangular element in the x and y directions, respectively. The resulting integral is given above as Equation [START_REF] Farhat | The discontinuous enrichment method[END_REF]. As indicated above the two integrations in the x 1 and x 2 directions can now be carried out in sequence. Now consider the simplest case, when p=0. The integral in the direction is simply

e iÿ dx = -ie iÿ ÿ + C (18)
and integrals of terms containing higher-order polynomials can be found by integrating by parts. A di culty arises since it is possible for ÿ to take the value 0. In this case the integral given in Equation ( 18) above must be evaluated using l'Hôpital's rule. If ÿ is small then numerical evaluations based on the Equation ( 18) would also be inaccurate. In such cases it is therefore necessary to adopt a series form for the exponential. This can be integrated term by term. A total of four special cases arises as indicated in Table I.

Table I.

1 2 3 4 ÿ = 0 ≈ 0 = 0 ≈ 0 = 0 = 0 ≈ 0 ≈ 0
When the absolute value of ÿ is below some set value, , the function e iÿ , is replaced by the truncated series

e iÿ ≈ N j=0 (iÿ ) j j! (19)
where N denotes the number of terms retained in the approximation. This can be adjusted, using the value , so that the truncation error in the series is of the order of machine accuracy.

The integral of the series in Equation ( 19) is well behaved as ÿ → 0, and so a seamless transition between the integrated form for the exponential and the series should be achieved.

A similar consideration applies in the case → 0, of course.

THEORY OF NUMERICAL INTEGRATION

There are many books and papers devoted to this topic, and we will make no attempt to cover the same ground. The necessary theory can be read in, for example, References [START_REF] Hamming | Numerical Methods for Scientists and Engineers[END_REF][START_REF] Davis | Numerical Integration (2nd edn)[END_REF] or [START_REF] Ralston | A First Course in Numerical Analysis[END_REF]. The simple procedure adopted here can be summarized as follows:

• Choose a set of integration absciss in two dimensions, distributed over the domain of the ÿnite element. In the present case, the integration absciss were chosen to be equally spaced in the and Á directions, and to include absciss at the ÿnite element vertices. For triangles the ranges were 06 61 -Á and 06Á61, and for the rectangle and quadrilateral -16 61 and -16Á61. Equal spacing was chosen because it was simple. § The number of points selected depends upon highest powers of and Á in the element. A bi-linear rectangular element matrix would have highest powers of 2 Á 2 , by deÿnition. This necessitates three integration points in each direction giving a total of nine points. In general if the highest power of and Á is n, then (n + 1) 2 points will be needed. A bi-quadratic element would need 25 integration points. • The Lagrange polynomials for each integration point are set up using the classical formula for such polynomials, [START_REF] Hamming | Numerical Methods for Scientists and Engineers[END_REF][START_REF] Ralston | A First Course in Numerical Analysis[END_REF], in both and Á directions.

L i ( )= n p=1 = i -p i -p L j (Á)= n q=1 = j Á -Á q Á j -Á q (20)
The polynomials for each point in the and Á directions are found and multiplied together. ¶ The next step is to integrate each polynomial over the domain of the element.

• Integrate analytically the product of the Lagrange polynomial and the wave functions over the domain of the ÿnite element, as seen in Equations ( 15) and ( 16) for the rectangle and triangle, respectively, to give the weights. § It may be possible to select optimal locations for the integration points, as in Gauss-Legendre integration. However, this would be almost impossibly di cult and the optimal locations would be di erent for every value of ÿ and .

There is a considerable literature on optimal integration absciss which cannot be explored here. See, for example References [START_REF] Hamming | Numerical Methods for Scientists and Engineers[END_REF][START_REF] Davis | Numerical Integration (2nd edn)[END_REF][START_REF] Ralston | A First Course in Numerical Analysis[END_REF]]. ¶ This is for the rectangle and quadrilateral. The corresponding polynomials are also available for triangles [START_REF] Zienkiewicz | The Finite Element Method[END_REF].

All these operations are ideal for being performed by a computer algebra program. The integration by parts, in two variables, gives rise to a very large number of terms, which are di cult to evaluate by hand. But because they are routine and standardized they are ideal for computer algebra packages. The existence of special cases when ÿ → 0 and → 0 further complicates the manipulations. The actual manipulations were carried out using Maple and Mathematica.

A DESCRIPTION OF THE MAPLE CODE

A web page is available which gives all the Maple codes and fortran codes, used to generate the results in this paper, [START_REF] Bettess | Web pages containing software relating to the present paper[END_REF]. The and Á co-ordinates are easily set for each integration point A switch for waves or no waves enables the program to be used simply for Newton-Cotes integrations if wished. In the latter case the ÿ and are set to zero. This is a simple and useful check since results can be compared with tables.

In the Maple code the two Lagrange polynomials for each integration point, were set up using a Maple procedure lagpoly, which simply implemented the classical formula for such polynomials, expressions [START_REF] Herrera | Connectivity as an alternative to boundary integral equations[END_REF].

After forming the product of each pair of polynomials, it was multiplied by the appropriate harmonic function and integrated analytically between the limits -1 and +1 in the and Á directions, using Maple. This value is then the weight for that integration point. The expressions for the weights in terms of the variables, a, b, ÿ and , are output as Fortran expressions. Maple can automatically generate Fortran or C code to generate these weights. The Fortran option is used here, but the change to C is tri ing.

The Fortran output is organized to be in the form of a subroutine, which can return the weights, on being supplied with the wavenumber, wave directions and element dimensions. This can be compiled directly by standard Fortran compilers. This would be called for a range of values of ÿ and . These in turn would depend upon the values of wavenumber, k, and wave directions  j .

It might be thought that an integration scheme which has to re-calculate the integration weights for every pair of wave directions would be ine cient. However, for short waves the expense of calculating and storing a large range of integration weights is small compared with the expense of the computations which must be carried out for all terms in the element matrix at each Gauss-Legendre point, in a numerically integrated element. The number of computations is also completely independent of the wavenumber.

INTEGRATION SCHEME LOGIC

Unlike conventional integration schemes, in which the same integration weights are used for all the terms evaluated at a given integration point, in this case each pair of wave directions requires a di erent, complex, weight. There is a signiÿcant overhead of storage and computation in evaluating and storing these integration weights. However, it is still economical compared with conventional Gauss-Legendre integration.

The procedure is as follows. For the ÿnite element all the directions at each node are assembled in a list. A simple example in which nodes 1, 2, 3 and 4 have 3, 2, 5 and 3 directions respectively, is shown below:

• Node 1 Â 1 1 , Â 1 2 , Â 1 3 • Node 2 Â 2 1 , Â 2 2 • Node 3 Â 3 1 , Â 3 2 , Â 3 3 , Â 3 4 , Â 3 5 • Node 4 Â 4 1 , Â 4 2 , Â 4 3
where  i j denotes the jth wave direction at the ith node. Let the total number of wave directions be M . In the above example there is a total of 13 wave directions. For each pair of directions, that is for M × (M + 1)=2 cases the integration weights are evaluated by a call to a fortran subroutine. In the example above there are 13 directions and hence 13 × 14=2=91 cases. The active arguments are the dimensions of the rectangular elements, a and b, and the exponents F and G, which are evaluated from Equation ( 9) from the wave number and wave directions. The integration weights are stored in a large array of dimension M × M × (n + 1) × (n + 1), where n is the highest power of and Á encountered. 5 is an appropriate value, if bi-quadratic elements are used. The weights are evaluated prior to the main integration loops.

Each integration loop is more complex than the conventional Gauss-Legendre-type integration, because as well as the main loops over all the integration points in the and Á directions, it is also necessary to have inner loops over all wave directions. However, the total number of multiplications of terms to evaluate the weighted residual is no larger, it is simply partitioned into smaller sections which use di erent weights. Moreover, some terms in the shape function and shape function derivatives are now present by implication, and not explicitly, so that the formation of the terms corresponding to a single integration point requires less e ort than for Gauss-Legendre integration. And, of course, there are, in general, far fewer integration points. In very general terms the logic is as follows:

For all x integration points do For all y integration points do For all i wave directions do For all j wave directions do Form contribution to element matrix at this integration point for these directions Multiply by weight for this point and these directions end do end do end do end do

EXAMPLE COMPARISONS

Consider a special wave element with 36 wave directions at each node and 9 nodes, using 100 × 100 Gauss Legendre integration. At each Gauss point three inner products of shape functions and shape function derivatives have to be formed to form the contribution to the element matrix. The size of the inner products is number of nodes multiplied by the number of wave directions, that is 36 × 9=324. Hence the number of oating point multiplications is 100 × 100 × 3 × 324 ≈ 10 000 000

In the case of 5 × 5 set of integration points, the total number of oating point multiplications is

5 × 5 × 3 × 324 ≈ 25 000
So provided the number of operations in evaluating the integration weights is less than 9 × 10 7 , the forming of the integration weights is worthwhile. That is we can a ord up to ÿve million oating point multiplications in evaluating the weights for each integration point. With 36 possible directions and 36 × 37=2 ≈ 670 combinations, even 500 oating point multiplications for each of these, is still worth while. In practice it is often possible to remain well below that ÿgure.

NUMERICAL RESULTS FOR RECTANGULAR ELEMENT

A program was written which would generate the element matrices using the special integration rules developed here, or many conventional Gauss-Legendre integration points. The element matrices obtained in this way were compared with each other. It was expected that the special integration rule would generate the element matrix accurately, irrespective of the wave number, k, whereas the Gauss-Legendre scheme would only be accurate if a su ciently large number of integration points were used. This proved to be the case.

The comparison between the semi-analytical integration and the Gauss-Legendre numerical integration was carried out as follows. The squares of the absolute values of the di erences were summed for every term in the two matrices and then divided by the sum of the squares of the absolute values of the terms in the matrix obtained by semi-analytical integration.

e = n i=1 n j=1 {|K(an) ij -K(gl) ij |} 2 n i=1 n j=1 |K(an) ij | 2 ( 21
)
where K is the element matrix, n is the total number of degrees of freedom in the element, (an) denotes the semi-analytical integration, (gl) denotes the Gauss-Legendre integration, and e is the root mean square error. This error is plotted in Figure 1.

The element was square, with length of side 2h. The program was run for a range of kh values between 4 and 140, and for a range of conventional Gauss-Legendre integration points between 1 and 120. A value of kh=140 corresponded to a wavelength, , of approximately =2 h=140 ≈ 0:045h This represents roughly 2=0:045 ≈ 45 wavelengths in the element.

Since the semi-analytical integration should be independent of the wavelength, (apart from the issue of rounding errors at very high values of k), we can predict the behaviour of the results. We should expect the di erence between the semi-analytical integrated value for the element matrix, and the numerically integrated value to be small for small values of k. As k increases we should expect the di erence, or error, to increase, when the Gauss-Legendre scheme no longer has su cient integration points to resolve the wave details. This is exactly what happens.

Figure 1 shows a plot of the error against the wavenumber, k, for a range of numbers of integration points. The semi-analytical and numerical errors increase at a value of k, which increases with the number of integration points. For very low numbers of integration points, between 1 and 10, the element matrix is not resolved at all with any acceptable accuracy. For numbers of points greater than 10, the error is around 10 -7 to 10 -6 , until the wavenumber increases to a value at which the number of integration points cannot resolve the wave details. Then as the wave number increases there is a rapid rise in the RMS error, which then plateaus at absurd and unacceptable values of about 10 3 . For example this transition takes place for 100 integration points, for k just below 100. For the results shown, four directions per node were used. Although the results are not shown, the program was also run for a range of di erent wave directions, and also for the special cases of F ≈ 0, G ≈ 0 and F ≈ G ≈ 0. (See Section 2.) No signiÿcant di erences in the results were observed.

TESTS AND TIMING RESULTS FOR RECTANGULAR ELEMENT

The two programs were used to evaluate the element matrices. Of course this is a subjective comparison, since one type of element may be coded more e ciently than the other. The speed-ups varied with the wave length, as shown in Figure 2. The time taken to integrate the element matrix semi-analytically has been found, using timing routines. This is denoted Figure 2. Ratio of analytical to numerical times for element matrix integration', R = T sa =T gl , as a function of number of integration points.

as T sa . The time to integrate the element matrix numerically has been similarly denoted T gl .

The ratio of these two times has been termed, for brevity, 'Ratio of analytical to numerical times for element matrix integration', R=T sa =T gl . In Figure 2 this time is plotted as a function of the number of Gauss-Legendre integration points in and Á directions. The plot in Figure 2 demonstrates that the semi-analytical integration gives a considerable beneÿt, especially for shorter wave lengths. The break-even point is for 10 × 10 Gauss-Legendre integration, which corresponds, roughly, to one wavelength in each direction in the special ÿnite element. For 120 × 120 integration, or 12 wavelengths in each direction, the advantage of the semi-analytical integration is a speed-up factor of about 120. This is the sort of zone in which the advantages of the special wave ÿnite elements are particularly felt. It should also be noted that no special e orts have been made to 'tweak' the integration code produced by Maple to be optimal. In general, although the code produced by Maple is adequate, it is not particularly e cient. Variables are often deÿned repeatedly. Also internal symmetries in the calculation of the integration weights have not been exploited. For these, and other reasons, it is felt that the speed-ups shown, are the minimum which can be expected from such techniques. (Fortran code produced by Mathematica was much worse than that produced by Maple.)

TRIANGULAR FINITE ELEMENTS

The integral for the weights for integration over the triangular ÿnite element is given above as Equation [START_REF] Ortiz | An improved partition of unity ÿnite element model for di raction problems[END_REF]. and Á can be identiÿed with the area co-ordinates L 2 and L 3 of the triangle, respectively, as is well known, and is shown in Figure 3.

L 1 =A 1 =A L 2 =A 2 =A L 3 =A 3 =A L 1 + L 2 + L 3 =1 ≡ L 2 Á ≡ L 3 ( 22
)
Because of the limits of integration for the triangle, another special case arises corresponding to that mentioned after Equation [START_REF] Sã Anchez-Sesma | A boundary method for elastic wave di raction: application to scattering of SH waves by surface irregularities[END_REF]. The case of ÿ ≈ must be considered. Terms of the Figure 3. Geometry of the triangular ÿnite elements, and the numerical integration points.

Table II.

1 2 3 4 5
F = 0 = 0 = 0 ≈ 0 ≈ 0 G = 0 = 0 ≈ 0 = 0 ≈ 0 F -G = 0 ≈ 0 = 0 = 0 ≈ 0
form e i(ÿ-)Á must then be replaced by the equivalent series. This gives a total of ÿve special cases, listed in Table II.

The polynomials for the triangle must be evaluated in terms of local area, or , Á coordinates, using expressions which are in standard textbooks [START_REF] Zienkiewicz | The Finite Element Method[END_REF], and which will not be repeated here. The integration points are chosen to be regularly spaced in and Á. The simplest formula thus has 1 integration point, at the centre of the triangle (although this would be of little practical interest), the second has three integration points, at the three vertices, the third has in addition the midside points, and so on. The layouts are sketched in Figure 3.

QUADRILATERAL FINITE ELEMENTS

In the case of the bi-linear quadrilateral the Jacobian of the mapping from local to global coordinates is no longer constant. The integrations will therefore only be approximate. However, it is possible to follow the same procedures as before. An additional term to those of Equations ( 11) is introduced,

t =k( ên + êj ) • d t (23)
After some manipulation and the assumption of a constant Jacobian the expressions for the weights become

w pq (ÿ; )= +1 -1 +1 -1
e iÿ e i Á e i Á L ' d dÁ [START_REF] Davis | Numerical Integration (2nd edn)[END_REF] In the integral of Equation ( 24), the di culties arise from the presence of the Á term. However, the integrals, while complicated, can be carried out. In the quadrilateral, there are three constants, ÿ, and , and a total of 12 special cases arise, corresponding to these values being zero or non-zero, and to ÿ or being approximately equal to , which again causes a problem. When is non-zero, the weights involve the exponential integral, Ei(x). This function is not supplied in Fortran, and so it was evaluated in our code in two separate ways. The ÿrst method was using the code from Numerical Recipes [START_REF] Press | Numerical Recipes in Fortran[END_REF]. The second was by using code written by Bulirsch [START_REF] Bulirsch | Numerical calculation of the sine, cosine, and fresnel integrals[END_REF] and modiÿed by Clark [START_REF] Clark | The ÿnite element method applied to Stokes wave di raction[END_REF]. This code uses Chebychev coe cients to evaluate the Cosine and Sine integrals. Polynomials are evaluated using Clenshaw's algorithm.

The values for Ei(x) obtained from the two codes were checked against each other and against values evaluated to arbitrary precision by Maple. Because 12 separate special cases arise, details are given elsewhere [START_REF] Sugimoto | A numerical integration scheme for special quadrilateral ÿnite elements for the Helmholtz equation[END_REF].

11. RESULTS

Evanescent modes

In order to test the rectangular element integration schemes on a realistic and non-trivial problem the so-called evanescent mode problem was solved in the interior of a rectangular domain. While the comparisons given in Figure 1 are useful, because of the ill-conditioned nature of the matrices generated by these elements it is essential to test the element matrices on a real problem. The solution to this problem is given by Morse and Feshback [START_REF] Morse | Methods of Mathematical Physics[END_REF], who demonstrate that the Helmholtz equation can be separated into two equations, so that Equation (1) becomes 1 X

d 2 X dx 2 + 1 Y d 2 Y dy 2 + k 2 =0 ( 25
)
where k is the wavenumber, given by k =2 = , where is the wavelength, and =X (x)Y (y).

After some manipulation this leads to a solution of the form

= exp [i x + y 2 -k 2 ] ( 26
)
where is an arbitrary constant. For values of smaller than k, the conventional plane wave propagating in a direction deÿned by the two x and y components is obtained. But if ¿k, the behaviour in the y direction is exponential, and the apparent wave length in the x direction is given by 2 = , and is therefore less than the true wavelength, given by 2 =k. Since the solution in this case has a di erent wavelength from that assumed in the special element shape functions it is a searching test of the approach. The solution can be further generalized by rotating it through an angle ÿ. The most general form is therefore where in the usual notation, with ÿ being the angle of rotation of the wave ÿeld

=a 0 exp (i s) exp(t 2 -k 2 ) (27)
s=x cos ÿ + y sin ÿ; t = -x sin ÿ + y cos ÿ (28)
The boundary condition applied on all four boundaries is the Robin boundary condition [START_REF] Laghrouche | Short wave modelling using special ÿnite elements[END_REF] @ @n + ik =g [START_REF] Clark | The ÿnite element method applied to Stokes wave di raction[END_REF] where g is the known solution, derived from Equations ( 28) and ( 29) above, and n is the outward normal.

For the numerical solutions, the domain is 06x612, 06y61:2. The problem is ÿrst solved for the parameters k =1, =1:5 and ÿ =30 • . For this case a mesh of 4 × 4 equal sized rectangular elements were used, with 8 wave directions at each node. Figure 4 shows the real part of the analytical solution for the evanescent mode. The real and imaginary potential along the centreline of the domain, y =0:6 is plotted in Figure 5, for analytical, and the numerically integrated and the semi-analytically integrated rectangular elements. Figure 6 shows the errors in the ÿnite element solutions along the same line.

The same evanescent modes problem is next solved for the parameters k =3, =5 and ÿ =10 • . The same element mesh was used as before but with 24 wave directions at each node. Figure 7 shows the real part of the analytical solution for the evanescent mode. For the numerical solutions Robin boundary conditions are used. The real and imaginary potential along the centreline of the domain, y =0:6 is plotted in Figure 8, for analytical, and the numerically integrated and the semi-analytically integrated rectangular elements. Figure 9 shows the errors in the ÿnite element solutions along the same line. The results are encouraging. Aside from the generally acceptable accuracy of the results, they demonstrate that the results from the numerically integrated elements and the semi-analytically integrated elements are virtually identical. This gives conÿdence in the scheme for the rectangular element.

Circular cylinder di raction

The problem considered consists of a rigid circular cylinder of radius, a=1. The ÿnite element mesh extends from r=a=1 to 5. The mesh is regular, with 36 sectors, each subtending 10 • . Each sector is modelled using 8 linear triangular elements. (Linear in the sense that the underlying shape function is linear.) The wavenumber, k is chosen so that ka=4. The mesh is shown in Figure 10. Figure 11 shows the results for plane waves di racted by the cylinder, for ka=4, where k is the wave number and a is the cylinder radius. All the waves considered are incident from the negative x direction, that is from  =180 • and r → ∞. They progress towards  =0 • and r → ∞. All incident waves have unit amplitude. The numerically integrated elements use a Gauss-Legendre scheme involving 8 integration points in two directions. (This unsymmetrical scheme is not entirely satisfactory, but we had no access to higher triangular integration schemes.) The elements on the boundary, used to implement the boundary conditions were integrated using 40 Gauss-Legendre integration points, and so the order of quadrature should not in uence the results, as it is so much higher than the minimum requirement. Plane damper boundary conditions were used, since they are simple to implement, and the present investigation is to determine the accuracy of the elements. The authors are aware that much more accurate techniques are available. Both numerically integrated and semianalytically integrated elements use 8 equally spaced wave directions, i.e. 0 • , 45 • ; : : : ; 315 • . Figure 12 shows the errors in the real and imaginary components of the complex wave potential as a function of the angle around the cylinder. The plots are of the di erence between the analytical expression and the numerical values. The results demonstrate that there are virtually no di erences between the values obtained by the analytically integrated and numerically integrated elements. Figure 13 shows the results for plane waves di racted by the cylinder, for ka=16, where k is the wave number and a is the cylinder radius. The numerically integrated elements use a Gauss-Legendre scheme involving 24 integration points in two directions. The same arrangement as before served for boundary elements and plane damper boundary conditions were again used. Both numerically integrated and semi-analytically integrated elements use 18 equally spaced wave directions, for this shorter wave problem i.e. 0 • ; 20 • : : : 340 • . Figure 14 shows the errors in the real and imaginary components of the complex wave potential as a function of the angle around the cylinder. The plots are of the di erence between the analytical expression and the numerical values. The results demonstrate that there are virtually no di erences between the values obtained by the analytically integrated and numerically integrated elements. For this single example the execution times for the formation of the element matrices and the solution of the ÿnite element equations are shown in Table III. The computations were carried out on a standard PC. In the case of the numerically integrated 24 × 24 integration point elements, the time to form the element matrices, was 899 s. That for the semi-analytically integrated elements was 60 s. This latter time was about the same as the system matrix solution time. Too much should not be read into one set of ÿgures, but they do indicate that the time for the formation of the element matrices can be brought broadly into line with the time for the solution of the system matrix, which indicates that the method should be competitive. The ÿgure for the semi-analytical integrations should be independent of wave length.

If the criterion of 10 nodes per wavelength, for wave modelling, is used a regular mesh would require 101 859 nodes, and an unstructured mesh 48 892 nodes. This assumes that pollution would not play a role, which might well be optimistic in a problem with over 12 wavelengths in the radial direction. In the present model, with 180 nodes, and 18 directions per node, the total number of degrees of freedom is 3240. This gives a factor of reduction of number of degrees of freedom for the unstructured mesh of roughly 15, and for the structured mesh roughly 31. It is likely that the number of nodes in the present mesh could be reduced if an iso-parametric version of these elements (which could model more accurately the curved cylinder), were developed, since the nodal spacing circumferentially is much closer than in the radial direction.

CONCLUSIONS

A method for integrating semi-analytically the element matrices arising from the new schemes of special ÿnite elements for the solution of Helmholtz equation in two dimensions has been presented. The results show that the integration schemes give results which are hardly distinguishable from those obtained using very large numbers of conventional Gauss-Legendre integration points. As expected the element matrix integrations appear to be independent of wave number, as shown by testing. For larger wavenumbers the semi-analytical integrations lead to large savings in execution times, by factors which increase with increasing wavenumber. Preliminary results for some classical wave problems demonstrate that these savings seem to be achieved without any loss of accuracy. At any rate for the elements of simpler geometry, semi-analytical integration seems to be a very promising way forward. There remain the challenges of higher-order parametric elements and the extension to three dimensions. In three dimensions the beneÿts of this approach should be even more marked, if it can be got to work. As the semi-analytical integration codes are still experimental it is likely that they could be improved upon. The other approach, that of Ortiz [START_REF] Ortiz | An improved partition of unity ÿnite element model for di raction problems[END_REF], could well be even more e cient. Time and experience will tell which method is the most e ective, but it would be premature to make a judgement at the moment.

The PostScript ÿle containing this paper, and the Maple codes to determine the integration weights, the corresponding generated Fortran integration weight codes and the test programs which generated the element matrix comparisons shown in Section 9, are all freely available on the web site [START_REF] Bettess | Web pages containing software relating to the present paper[END_REF].

Figure 1 .

 1 Figure 1. RMS errors in the element matrix, as a function of wavenumber and number of integration points.

Figure 4 .

 4 Figure 4. Real part of the rotated evanescent mode, = exp(i x + y √ 2k 2 ), for k =1, =1:5 and ÿ =30 • , in the domain 06x612; 06y61:2.

Figure 5 .

 5 Figure 5. Real and imaginary potentials for the rotated evanescent mode, = exp(i x + y √ 2k 2 ), for k =1, =1:5 and ÿ =30 • , along the centreline of the domain, y =0:6, showing the analytical, numerically integrated and semi-analytical results.

Figure 6 .√ 2 -

 62 Figure 6. Real and imaginary errors in potentials of the rotated evanescent mode, = exp (i x+ y √ 2k 2 ), for k =1, =1:5 and ÿ =30 • , along the centreline of the domain, y =0:6, showing the numerically integrated and semi-analytical results.

Figure 7 .

 7 Figure 7. Real part of the rotated evanescent mode = exp(i x + y √ 2k 2 ), for k =3, = 5 and ÿ =10 • , in the domain 06x612; 06y61:2.

Figure 8 .

 8 Figure 8. Real and imaginary potentials for the rotated evanescent mode = exp(i x + y √ 2k 2 ), for k =3, = 5 and ÿ =10 • , along the centreline of the domain, y =0:6, showing the analytical, numerically integrated and semi-analytical results.

Figure 9 .

 9 Figure 9. Real and imaginary errors in rotated evanescent mode = exp(i x + y √ 2k 2 ), for k =3, = 5 and ÿ =10 • , potentials along the centreline of the domain, y =0:6, showing the numerically integrated and semi-analytical results.

Figure 10 .

 10 Figure 10. The mesh of 'linear' triangular ÿnite elements used for the cylinder wave di raction problem.Inner radius 1, outer radius 5.

Figure 11 .

 11 Figure 11. Real and imaginary potentials along the circumference of the di racting cylinder, showing analytical, numerically integrated and semi-analytical results for ka =4.

Figure 12 .

 12 Figure 12. Real and imaginary errors in potentials along the circumference of the di racting cylinder, for numerically integrated and semi-analytical elements, for ka =4.

Figure 13 .

 13 Figure 13. Real and imaginary potentials along the circumference of the di racting cylinder, showing analytical, numerically integrated and semi-analytical results for ka = 16.

Figure 14 .

 14 Figure 14. Real and imaginary errors in potentials along the circumference of the di racting cylinder, for numerically integrated and semi-analytical elements, for ka = 16.

Table III .

 III

		Element	System
		matrix	matrix
		formation	solution
	Numerical elements	899	61
	Semi-analytical elements	60	61
	Note: All times in seconds.

ACKNOWLEDGEMENTS

This work has been funded by the EPSRC, under grants GR=K81652, GR=M41025 and GR=M41018. We are very grateful to EPSRC for this support for Omar Laghrouche and Rie Sugimoto. Peter Bettess is grateful to the University of Durham, for its support through a Sir Derman Christopherson Fellowship, for the academic year 1999-2000 and to the Ecole Centrale de Nantes and its director, Pierre Vaussy, for the resources to work in Nantes in the Summer of 1999. Peter Bettess is also grateful to the EPSRC Senior Research Fellowship committee, chaired by Professor Richard Brook, which is funding this work, through EPSRC Grant Number: SF/000169.

We are also grateful to Dr P. Ortiz and Dr E. Perrey-Debain for very useful discussions.