Pavel Livotov

Arun Mas'udah

Chandra Prasad

Sekaran

On the Efficiency of TRIZ Application for Process Intensification in Process Engineering

Keywords: TRIZ Efficiency, Process Engineering, Process Intensification

In recent years, the application of TRIZ methodology in the process engineering has been found promising to develop comprehensive inventive solution concepts for process intensification (PI). However, the effectiveness of TRIZ for PI is not measured or estimated. The paper describes an approach to evaluate the efficiency of TRIZ application in process intensification by comparing six case studies in the field of chemical, pharmaceutical, ceramic, and mineral industries. In each case study, TRIZ workshops with the teams of researchers and engineers has been performed to analyze initial complex problem situation, to identify problems, to generate new ideas, and to create solution concepts. The analysis of the workshop outcomes estimates fulfilment of the PI-goals, impact of secondary problems, variety and efficiency of ideas and solution concepts. In addition to the observed positive effect of TRIZ application, the most effective inventive principles for process engineering have been identified.

Introduction

Innovation in process engineering is becoming more challenging in developing new processes or in transforming conventional chemical processes into more economical, productive, and environmental friendly. Process Intensification (PI) is a knowledgebased innovation methodology evolved over the last three decades mainly for gas-liquid systems, is focusing now on processes involving solids handling [START_REF] Wang | A review of process intensification applied to solids handling[END_REF]. Besides its basic concept of equipment miniaturization, the new PI equipment and methods enable to substitute common batch processes into faster and safer continuous processes with higher efficiency, productivity, and reduced costs [START_REF] Wang | A review of process intensification applied to solids handling[END_REF].

PI is expected not only to improve processes and resulting products, but also able to bring a high value profit through innovation. However, due to the secondary problems, the implementation of the novel PI apparatus and methods is not as simple as perceived. The use of systematic creativity approaches for PI seems to be an opportunity to faster overcome the secondary problems. Therefore, in the last few years the analytical and inventive tools of the theory of inventive problem solving TRIZ [START_REF]VDI Standard 4521: Inventive Problem Solving with TRIZ. Fundamentals and Definitions[END_REF] have been applied for intensification of various processes to ensure PI goals are achieved. As outlined in [START_REF] Casner | TRIZ-based approach for process intensification and problem solving in process engineering: concepts and research agenda[END_REF], TRIZ helps to identify secondary problems of PI and limits these negative side effects. The benefits of TRIZ approach above conventional creativity methods in practice are reported in several interdisciplinary domains [START_REF] Hua | Integration TRIZ with problem-solving tools: a literature review from 1995 to 2006[END_REF][START_REF] Ilevbare | A review of TRIZ, and its benefits and challenges in practice[END_REF][START_REF] Chechurin | Understanding TRIZ through the review of top cited publications[END_REF], as well as in the field of process engineering [START_REF] Casner | Advanced innovation design approach for process engineering[END_REF][START_REF] Pokhrel | Adaptation of TRIZ contradiction matrix for solving problems in process engineering[END_REF] and in particular of chemical processes [START_REF] Kim | Application of TRIZ creativity intensification approach to chemical process safety[END_REF]. The authors of [START_REF] Pokhrel | Adaptation of TRIZ contradiction matrix for solving problems in process engineering[END_REF] adapt TRIZ contradiction matrix to resolve contradictions related to process engineering problems. Their research proposes new characteristics and inventive principles for TRIZ contradiction matrix to counter problems in chemical process industries. Another work introduces TRIZ methodology and framework for problem solving and forecasting of product development in chemical engineering [START_REF] Abdul Rahim | TRIZ methodology for applied chemical engineering: A case study of new product development[END_REF]. Furthermore, the authors of [START_REF] Abramov | TRIZ-based approach for accelerating innovation in chemical engineering[END_REF] demonstrate that TRIZ-based approach is well-suited for accelerating innovation in chemical engineering. Additionally, TRIZ application has been modified and enhanced to the design of inherently safer chemical processes [START_REF] Srinivasan | Application of the TRIZ creativity enhancement approach to design of inherently safer chemical processes[END_REF]. The coupling of TRIZ with the Case-Based Reasoning in chemical engineering is proposed in [START_REF] Cortes Robles | Case-based reasoning and TRIZ: A coupling for innovative conception in Chemical Engineering[END_REF], and the advantages of linking TRIZ and Process Intensification are disclosed in [START_REF] Livotov | Systematic Innovation in Process Engineering: Linking TRIZ and Process Intensification[END_REF]. Evidently, TRIZ helps engineers to create inventive solutions and to deliver radical improvements, but it still shows lower application level in industrial practice, ranking for example with 14% in fifth place of the top five creativity methods supporting innovation process in German companies [START_REF] Livotov | Advanced Innovation Design Approach: towards Integration of TRIZ Methodology into Innovation Design Process[END_REF]. Therefore, the objective evaluation of the TRIZ effectiveness for process engineering remains a significant research issue.

This paper presents an effort to evaluate the efficiency of TRIZ application in process intensification involving solids handling through six case studies assessment in the field of chemical, pharmaceutical, ceramic, and mineral industries as shown in Table 1. To reach the research objectives, in each case study a one or two-days TRIZ workshop was prepared, carried-out and evaluated with the team of researchers and engineers. Each TRIZ workshop emphases on improving abilities in systematic problem definition and ideation. It starts with the comprehensive analysis of the initial problem situation and ends with the creation of alternative or complimentary solution concepts for the PI case studies. The objective measures of ideation effectiveness defined in [START_REF] Shah | Metrics for measuring ideation effectiveness[END_REF] such as variety, quantity and quality of proposed novel ideas and solution concepts are documented and evaluated. The corresponding most effective TRIZ inventive principles and sub-principles frequently used in creation of PI solution concepts are identified. Additionally, a few months later review is conducted for each case study to observe and evaluate the positive effect of TRIZ application in the practice.

2

Case Study Overview

CS1 -Separation of Ceramic-Metal Powders

The case study CS1 aims to intensify the powder classification system for separation of ceramic-metal powders produced by high energy ball milling. The system should be able to divide the input powder by mass or size without disposable parts, in continuous and without human intervention for powder handling. Even though the existing system of air classification and sieving system is simple and flexible for powders classification, it still has disadvantages on discontinuous process, generation of fine phase in milling process and generation of environmental contaminant particles.

CS2 -Dry granulation of Ceramic Powders

High shear mixer granulator (HSMG) is an established technology for dry route of ceramic powders granulation with lower energy and water consumption which is not shown in wet route processing. However, dry granulation contains many constraints regarding final granulated product such as higher moisture content, bimodal size distribution and low productivity. Therefore, the main challenge of case study is to produce granulated powders with adequate physical and chemical properties through the dry route with the same properties as the ones obtained through the wet route.

CS3 -Metal Ore Beneficiation

In mineral beneficiation processes, on-line mineral analysis based on Raman spectroscopy monitors mineral concentrations at selected locations along the process line. Routine operation of the system is easy and can be carried out by persons who have no indepth knowledge about the fundamentals of Raman spectroscopy. Nevertheless, undesired measurement deviations caused by the presence of dark and different minerals, varying particle size of incoming ore, varying solids content and poor representability of the sampling have to be minimized.

CS4 -Wet Granulation of Pharmaceutical Powders

The objective for this case study is to perform wet granulation of pharmaceutical powders in a single shift which is currently carried out in four shifts. This method would reduce the number of granulations per batch, and in consequence, decrease the process time per batch. The main task of the new process is to obtain stable homogeneous granules in a wet high-speed mixer, with set dissolution and other analytical specifications achievable in the current process.

CS5 -Drying of Pharmaceutical Powders

Drying is one of the heart processes in granulation of pharmaceutical powders. Spiral Flash Dryer (SFD) is a swirling fluidized bed reactor [START_REF] Wang | A review of process intensification applied to solids handling[END_REF] and can be considered as an alternative to conventional flash dryers and other fluid bed applications. The SFD is designed to dry granules with a minimal amount of drying air and providing intrinsic separation of dry particles from wet particles, which are recirculated and dried further.

In addition, the operating principle is similar with an expanded fluid bed, with a very short residence time, maintaining a fast-drying effect due to the high-induced turbulence. The intensification goal of this case study is preventing undesired properties of SFD such as moisture deviation, generation of fine dry powders and sticky product over the equipment (fouling of solids).

CS6 -Mixing of Chemical Reagents

The CoFlore® Agitated Tube Reactor is a general purpose dynamically agitated chemical reactor, and therefore offers a range of applications as broad as that of existing batch reactor technology [START_REF] Wang | A review of process intensification applied to solids handling[END_REF]. It can handle a large variety of reaction schemes, sequences, processes, and phase combinations over a wide range of reaction and residence times. Additionally, it also gives high level of mixing, reduces the required maintenance, and improves operational life of the reactor, allowing for easy disassembly and cleaning. However, the system should be enhanced to give the best possible mixing characteristics, to increase uniformity of solid distribution, to enable scalability of reactor, and to avoid fouling of solids.

Research Approach

Concerning inventive problem solving for process intensification, TRIZ offers a systematic direction to figure out ideas and solutions for developing new technology or improving the existing one. Even though TRIZ does not give directly an applicable solution, it helps to generate novel ideas faster and provides a basis for selecting the best ideas [START_REF] Cortes Robles | Case-based reasoning and TRIZ: A coupling for innovative conception in Chemical Engineering[END_REF]. According to the Advanced Innovation Design Approach (AIDA) for process engineering [START_REF] Casner | Advanced innovation design approach for process engineering[END_REF], the application of TRIZ follows a process with a series of phases:

I. Identification, analysis and ranking of partial problems. II.

Systematic idea generation with TRIZ inventive principles. III.

Creation of the solution concepts. IV.

Concepts optimization with elimination of the negative side effects.

In this research work, the first 3 process phases (I, II and III) were performed during TRIZ workshops in each of six case studies, with the process engineers and academia researchers, currently involved in the corresponding PI projects. Each workshop was carried out under the guidance of one TRIZ specialist as moderator within 1 or 2 days with 6…9 participants working in smaller teams of 2…3 persons. The relative duration of the single workshop phases was I -25%, II -50%, III -25% of the total time. The ideation phase II comprised one ideation session in the one-day workshops (CS1, CS3), and two ideation sessions in the two days workshops (CS2, CS4, CS5, CS6). The presented research assesses the efficiency of TRIZ application for problem solving, idea generation and concepts creation. Due to the time limit attached to the workshop duration, the process phase IV "Concepts optimization" was not a part of the workshops.

Phase I: Problems Identification, Analysis and Ranking

The approach starts with analyzing initial situation of process intensification using innovation situation questionnaire, followed by the comprehensive problem analysis including understanding primary function of equipment, operation, environment and working conditions for each case study. Function analysis and process mapping techniques [START_REF] Casner | Advanced innovation design approach for process engineering[END_REF] break down the complete production process into unit operations which results in identification of useful functions and undesired properties of unit operations. Finally, identification of the key negative effects or disadvantage of the analyzed system in accordance to the principles of Cause Effect Chain Analysis and Root-Conflict Analysis RCA+ [START_REF]VDI Standard 4521: Inventive Problem Solving with TRIZ. Fundamentals and Definitions[END_REF], helps workshop participants to identify the root causes of problems and to rank them accordingly the objectives of process intensification in each case study, as presented in Table 2. The problems were formulated in detail as (a) enhancement of insufficient positive function, (b) elimination of negative effect or (c) engineering or physical contradiction.

Phase II: Systematic Idea Generation with TRIZ

As a majority of workshop participants were not skilled in the TRIZ methodology, this phase of the workshop required a 45 minutes introduction to the TRIZ basics with a short training in application of 40 inventive principles. The enhanced version of 40 inventive principles with in total 160 sub-principles [START_REF] Livotov | Systematic Innovation in Process Engineering: Linking TRIZ and Process Intensification[END_REF] was applied in the workshops as a universal ideation tool convenient both for process engineers and researchers. The idea generation was also supported by solutions examples in chemical and process engineering known in TRIZ literature [START_REF] Grierson | 40 Principles -Chemical Illustrations[END_REF][START_REF] Hipple | 40 inventive principles with examples for chemical engineering[END_REF]. The application of the principles was performed for each partial problem separately in the order proposed in Table 3, whereby each idea generation phase was started with statistically strongest principles (group 1), optionally followed by the group 2 in case of design problems or group 3 for process optimization problems. All generated ideas were documented with a description and mandatory drawing and assigned to the inventive principles they were based on.

As a rule, different partial problems presented in the Table 2, were operated by different working teams. If more than one team worked on same partial problem, each team applied different TRIZ inventive principles to avoid similar ideas. The Fig. 1 shows statistics of inventive principles of three groups applied by the participants during TRIZ workshops for idea generation. The lowest amount of the inventive principles was used in one-day workshops with one ideation session (CS1 -8 principles; CS3 -11 principles). The number of principles used in two-days workshops with accordingly two ideation session (CS2, CS4, CS5, CS6) varies between 14 and 23. The ideation session duration was on average 3 hours including idea generation (1,5 hours) and idea presentation and discussion (1,5 hours). The identification of the most effective inventive principles and corresponding sub-principles applied by the participants for idea generation was one of the tasks in presented research.

Phase III: Concepts Creation

Idea generation phase provides solution ideas for the partial problems with different stages of concretization. To develop a new solution concept at least one solution idea must be selected for each partial problem. Thus, a robust solution concept delivers solutions for all partial problems. Workshop participants with their specific knowledge on processes, existing and new technologies are in position to select most promising ideas overcoming the partial problems and to combine complementary partial solutions into solution concepts. Due to the multi-objective aspect of the concept creation and optimization, more than one solution concept can be designed in this phase, including "best performance", "minimum cost" or "optimal performance to cost ratio" conceptual alternatives. Therefore, the creation of solution concepts in a situation comprising several problems remains one of the challenging phases in the workshops. For example, Table 4 compares briefly six solution concepts developed in case study 5 (CS5) from in total 52 ideas. As one can see, the concepts C1 -C5 contain solutions for all three partial problems, and the concept C6 offers solutions for the problems 1 and 3 only.

TRIZ Impact on Quantity and Novelty of proposed Ideas

The quantitative analysis of the case studies undoubtedly shows that TRIZ methodology can effectively support the process intensification tasks in the industry and with 234 novel ideas and 28 solution concepts for 15 partial problems consistently yields good results in the variety and novelty of ideas and solution concepts. It is important to note that in each case study all problem-solving attempts and ideas known prior to the workshops were documented in the initial situation questionnaire. Thus, the discussed ideation outcome of the workshops includes different ideas only, new to the initial situation. In other words, all 234 solution ideas can be considered as dissimilar and novel. Furthermore, the combination and adaptation of the selected novel ideas to the solution concepts leads to the patentable inventions and creation of new intellectual property.

As presented in Table 5, each case study (CS) can be served with several ideas and solution concepts: for instance, 17 ideas and 3 concepts in the CS1, 58 ideas and 8 concepts in the CS2, etc. The mean number of solution ideas proposed for each partial problem is 15,6. The average number of ideas generated in one ideation session (IS) by a team is 9,17 (SD=1,51). The mean value of dissimilar new ideas proposed by each workshop participant in one ideation session along all case studies is 3,12 (SD=0,52). However, a missing direct comparison with a control group of engineers working without TRIZ belongs to the limitation factors of the present analysis.

TRIZ Impact on Variety of proposed Ideas

In order to assess the variety of the ideas proposed in the workshops, every idea was assigned to the most appropriate knowledge domain. For this purpose, both problems and generated ideas were related to the eight engineering MATCEMIB fields and their interactions [START_REF] Belski | Eight Fields of MATCEMIB Help Students to Generate More Ideas[END_REF] applied in TRIZ: Mechanical, Acoustic, Thermal, Chemical, Electric, Magnetic, Intermolecular, Biological. For example, a uniformity problem caused by the surface effects of solid particles can be assigned to the intermolecular field, and a corresponding solution idea based on ultrasound treatment -to the acoustic field.

IDEAS FIELDS

As shown in Figure 2, the 15 partial problems presented in all case studies belong to following engineering fields: chemical problems 40%, mechanical problems 28%, intermolecular problems 28%, and thermal problems 4%. At the same time, the 234 ideas from all case studies can be assigned to all fields excepting biological. As a result, the majority of the problems could be solved within mechanical (55%), chemical (16%) and intermolecular (13%) knowledge domains in the process engineering.

The variety of solution ideas in relationship between the engineering domains of problems and proposed ideas is illustrated in Table 6. Thanks to the TRIZ inventive principles engineers were able to find solution ideas in 7 engineering domains for mechanical, chemical, and intermolecular problems. However, no ideas based on acoustic, electric, or magnetic fields were proposed for thermal problems.

Quality of proposed Ideas and Ideation Efficiency

Despite of high quantity and variety of the novel ideas proposed with TRIZ inventive principles, the industrial companies require technically feasible, economically affordable, and safe solutions. On that reason, only a smaller number of high quality ideas has been selected by the engineers for further implementation in the solution concepts taking into account the following criteria: idea value (how a partial problem is solved), implementation efforts, costs and technical risks. The number of such implementation ideas Id divided by the total number of the ideas Id t generated in one case study (see Table 5) can characterize the efficiency of an ideation technique. These efficiency values are presented for each case study in Figure 3. For example, in the case study CS1 (Id=6; Id t =17) only 35% of all generated ideas were considered as mature for further implementation and were applied for concepts creation. The mean value of TRIZ efficiency for all Process Intensification case studies can be calculated with the formula (1) and is equal 33,8%.

Thus, roughly only one third of the proposed ideas is estimated by the engineers as especially promising for process intensification. However, in the opinion of researchers many of ideas, which were not selected by engineers for implementation, possess a substantial potential for significant technological innovation in future.

𝐸

( 1 )

E Id -mean efficiency of ideation estimated on the base of all case studies Id i -number of implementation ideas in one case study used for concepts creation Id ti -total number of ideas generated in one case study n -number of case studies Fig. 3. Ideation efficiency with TRIZ inventive principles in the case studies CS1 -CS6, defined as the ratio of implementation ideas and total number of ideas in a case study.

Identification of most efficient Inventive Sub-principles

The analysis of the solution ideas and concepts created in six case studies has also resulted in identification of the most frequently used top 10 inventive sub-principles, as presented in Figure 4. These top 10 sub-principles can be generally considered as most efficient and recommended for the new development or optimization of PI equipment or methods in the field of processes involving solids handling. The identified sets of strongest sub-principles differ from the recommended group of statistically strongest inventive principles, presented in Table 3.

To the most often applied sub-principles belong Change concentration (35b), followed by Use electromagnetism (28a), and Reversed sequence (13b). It is also interesting to notice that top 10 sub-principles used in idea generation such as Periodic action (19a), Optimal performance (15a), Gaseous or liquid flows (29a), Miniaturization (17b) and Pre-arrange objects (10b) don't appear in top 10 sub-principles applied in solution concepts. Vice versa, the top 10 sub-principles Utilize harm (22a), Spheres and cylinders (14b), Different functions (3c), One step back from ideal (16a) and Rotary motion (14c) found in the solution concepts don't belong to the top 10 sub-principles in idea generation phase.

The TRIZ sub-principles mentioned above belong to the updated version of 40 inventive principles with in total 160 sub-principles presented in the previous research [START_REF] Livotov | Systematic Innovation in Process Engineering: Linking TRIZ and Process Intensification[END_REF]. The study [START_REF] Livotov | Systematic Innovation in Process Engineering: Linking TRIZ and Process Intensification[END_REF] analyses 155 Process Intensification technologies and 150 patent documents in the field of solid handling in the ceramic and pharmaceutical industries. It partially restructures and enhances the classical 40 inventive principles and extends them with additional 70 inventive sub-principles for the problems frequently encountered in process engineering. The identified sets of top 10 TRIZ inventive principles and sub-principles in the analyzed case studies show clear differences to the corresponding top 10 sets of frequently used TRIZ principles and sub-principles extracted from the 155 PI technologies show some differences and from 150 patents for intensification ceramic and pharmaceutical operations with solids in previous investigation [START_REF] Livotov | Systematic Innovation in Process Engineering: Linking TRIZ and Process Intensification[END_REF]. As illustrated in Figure 5, there are 9 inventive principles (No 1, 5, 6, 14, 18, 24, 28, 29 and 35) most frequently used in all three sources of analysis: I) Case studies, II) PI technologies, III) Patents. The comparison on the level of sub-principles delivers a higher resolution and depicts only 5 frequently used sub-principles: Spheres and cylinders (14b), Rotary motion (14c), Use electromagnetism (28a), Change concentration (35b) and Change temperature (35d). In general, the identified selection of principles and sub-principles in Fig. 5 can be recommended for problem solving in processes involving solids handling.

The top 10 inventive principles and sub-principles applied in the case studies as well as top 10 inventive principles identified in the PI technologies and patents in [START_REF] Livotov | Systematic Innovation in Process Engineering: Linking TRIZ and Process Intensification[END_REF] belong to the different groups of TRIZ principles recommended for ideation in Table 3. This statement is illustrated in the Fig. 6. Understandably, that 9 of 10 top principles (left diagram) and sub-principles (right diagram) applied in 234 case studies (CS) ideas come from the group 1 of statistically strongest principles as group 1 was primarily recommended for ideation in each workshop. However, the solution concepts in the case studies are based on principles from all three groups. More uniform distribution of top 10 principles over the groups 1, 2 and 3 can be observed for PI technologies and patents. Fig. 6. Distribution of top 10 inventive principles (left) and top 10 sub-principles (right) over 3 groups of principles recommended for ideation: Group 1: statistically strongest principles; Group 2: principles for design problems; Group 3: Principles for process engineering problems.

Efficiency of TRIZ Application for Process Intensification

A series of reviews has been conducted several months after the workshops to observe and assess the positive effect of TRIZ application in the case studies. The industrial teams of engineers or researchers who participated in the TRIZ workshops, evaluated the current state of their PI projects with different criteria, such costs, time expenditures, technical risks, innovation impact and TRIZ support, as presented in Table 8. For each criterion the following scale has been used: 100% -very high, 80% -high, 60% -middle, 40% -low, 20% -very low. The estimated supportive impact of TRIZ workshops in the case studies was with on average 80% high. The evaluation results of the case study 3 could not be made available to the public.

Conclusion and Outlook

The presented research assesses the efficiency of TRIZ support for process intensification (PI) applied to solids handling in six industrial case studies in the field of chemical, pharmaceutical, ceramic, and mineral processes. The positive experience with combination of TRIZ and PI gives us confidence to extend the findings regarding TRIZ efficiency to the other fields of process engineering.

The application of TRIZ as a systematic problem-solving method demonstrates a positive impact on faster achieving PI goals. Moreover, TRIZ allows to identify and solve secondary PI problems comprehensively with numerous and interdisciplinary solution concepts. TRIZ inventive principles improve the outcomes of idea generation regarding novelty, quality, quantity, and variety of ideas.

Furthermore, it is worth to mention the identified sets and application order of the strongest inventive principles (14, 35, 5, 28, 1, 29, 6, 18, 24, 13, …) and inventive subprinciples (35b, 28a, 14b, 14c, 35d, 13d, 22a, 36a, 3c, 40a, … ) which can be recommended for systematic and creative problem solving in processes involving solids handling.

The later review of the case studies progress shows that most engineers and researchers positively judge the TRIZ application in the PI practice. However, the research is still required to optimize and formalize the phase of concept creation as wells as to better identify inventive solution principles and sub-principles for other PI domains.

Group 2 : 3 :

 23 Principles for solving design problems (6) Universality, (5) Combining, (29) Pneumatic or hydraulic constructions, (30) Flexible shells or thin films, (7) Nesting / Integration, (8) Anti-weight, (4) Asymmetry, (40) Composite materials, (24) Mediator, (14) Spheroidality and Rotation, (23) Feedback and automation, (31) Porous materials, (25) Self-service. Group Principles for specific problems in Process Engineering (16) Partial or excessive action, (27) Disposability / Cheap short living objects, (20) Continuity of useful action, (32) Change colour, (21) Skipping / Rushing through, (11) Preventive measure / Cushion in advance, (33) Homogeneity, (22) Converting harm into benefit, (39) Inert environment, (37) Thermal expansion, (36) Phase transitions, (38) Strong oxidants, (34) Rejecting and regenerating parts, (12) Equipotentiality, (9) Prior counteraction of harm.

Fig. 1 .

 1 Fig. 1. Amount of inventive principles in corresponding groups applied in the case studies.

Fig. 2 .

 2 Fig. 2. Distribution of problems and ideas over the eight fields of MATCEMIB in all case studies.

Fig. 4 .

 4 Fig. 4. Inventive sub-principles most frequently encountered in ideas and solution concepts.

Fig. 5 .

 5 Fig. 5. Top 10 most frequently used TRIZ inventive principles (left) and sub-principles (right) in solution concepts of 6 case studies (I), 155 PI technologies (II) and 150 patent documents (III).

Table 1 .

 1 Six case studies of intensification of processes involving solid handling.

	Case study (CS)	Process description
	CS1	Separation of ceramic-metal powders
	CS2	Dry granulation of ceramic powders
	CS3	Metal ore beneficiation
	CS4	Granulation of pharmaceutical powders
	CS5	Drying of pharmaceutical powders
	CS6	Mixing of chemical reagents

Table 2 .

 2 List of partial problems with higher ranking in case studies.

	Case study	PI technology	Partial problems (Pp)
	CS1 -Separation	Elbow-Jet Air Classifier	1. Avoid discontinuity in classification of
	of ceramic-metal	(EJAC)	powders
	powders		2. Avoid fine phase powders while milling
	CS2 -Dry gran-	High Shear Mixer Gran-	1. Reduce moisture content of granules
	ulation of ce-	ulator (HSMG)	2. Avoid bimodal size distribution of gran-
	ramic powders		ules
			3. Increase productivity
	CS3 -Metal ore	Flash Flotation Cell -	1. Avoid dark and different minerals
	beneficiation	Raman spectroscopy	2. Reduce variability in size and solids con-
			tent of incoming ore
	CS4 -Granula-	Wet High-Speed Mixer 1. Achieve stability of product
	tion of pharma-		2. Achieve target dissolution of product
	ceutical powders
	CS5 -Drying of	Spiral Flash Dryer	1. Achieve desired final moisture content
	pharmaceutical	(SFD)	of granules
	powders		2. Avoid fine powder generation
			3. Avoid sticky product over equipment
	CS6 -Mixing of	CoFlore® Agitated	1. Increase uniformity of solid distribution
	chemical rea-	Tubular Reactor	2. Scalability of reactor
	gents	(CATR)	3. Connectors optimization

Table 3 .

 3 Recommended order for application of 40 inventive principles.

	Group 1:	(35) Transform physical and chemical properties, (10) Prior useful
	Statistically strongest	action, (1) Segmentation, (28) Replace mechanical working princi-
	principles [9]	ple, (2) Leaving out / Trimming, (15) Dynamism and adaptability,
		(19) Periodic action, (3) Local quality, (17) Shift to another dimen-
		sion, (13) Inversion, (18) Mechanical vibration, (26) Copying.

Table 4 .

 4 Amount of ideas selected for use in 6 solution concepts of the case study CS5.

		Partial prob-	Solution		Ideas selected for solution concepts C1 -C6
		lems (Pp)	ideas	C1	C2	C3	C4	C5	C6
		Pp1	31	3	3	1	1	2	4
		Pp2	9	2	1	1	1	1	-
		Pp3	12	2	1	3	1	3	1
		Total	52 ideas	7 ideas 5 ideas 5 ideas 3 ideas 6 ideas 5 ideas
	4	Discussion of Results

Table 5 .

 5 Number of ideas, solution concepts, ideation sessions and teams in the case studies.

	Case study	Partial problems	Solution ideas	Solution concepts	Ideation sessions (IS)	Number of teams (persons)	Ideas in one IS per team (person)
	CS1	3	17	3	1	2 (6)	8,5 (2,8)
	CS2	3	58	8	2	3 (9)	9,7 (3,2)
	CS3	2	24	4	1	2 (6)	12,0 (4,0)
	CS4	2	28	3	2	2 (6)	7,0 (2,3)
	CS5	3	52	6	2	3 (9)	8,7 (2,9)
	CS6	3	55	4	2	3 (8)	9,2 (3,4)
	Total	15 partial	234	28 con-	10 ses-	15 teams	Mean
		problems	ideas	cepts	sions	(44 persons)	9,2 (3,1)

Table 6 .

 6 Relationship between engineering domains of partial problems and ideas.

	Engineering domains of 15 problems Engineering domains of 234 ideas	Mechanical Thermal Chemical	Inter-molecular
	Mechanical	x	x	x	x
	Acoustic	x		x	x
	T h e r m a l	x	x	x	x
	Chemical	x	x	x	x
	Electric	x		x	x
	Magnetic	x		x	x
	Intermolecular	x	x	x	x

II. PI techn. 36, 2 III. Patents 40, 33 28 1 14, 35 5 29, 6, 18 24

	40% 50%	35%	36%	29%	39%	33%	31%
	30%
	20%
	10%
	0%
			I. Case studies 13, 3, 16, 22, 2		I. Case studies 13d, 22a, 3c, 35c 16a, 3d
					28a, 14b 14c		35b
					II. PI techn. 36a, 2e, 29e 6b, 35a, 5a	35d	III. Patents 24a, 20a 40a, 40c, 33a 6a, 29a, 31a

Table 8 .

 8 Evaluation of the current state of PI design in case studies.

		CS 1	CS2	CS3	CS4	CS5	CS6
	1. Costs, level of investments	40%	60%	-	80%	80%	60%
	2. Time expenditures for imple-	60%	80%	-	60%	-	80%
	mentation of technology
	3. Level of anticipated technical	60%	80%	-	40%	80%	40%
	risks in future
	4. Innovation impact of new tech-	80%	80%	-	100% 100% 100%
	nology in terms of PI-goals
	5. Supportive impact of TRIZ	80%	60%	-	100%	80%	-

Acknowledgments. The authors thank the European Commission for supporting their work as part of the research project "Intensified by Design® platform for the intensification of processes involving solids handling" under H2020 SPIRE programme.


Anass Serhani

Ghislain Haine

Denis Matignon

Anisotropic heterogeneous n-D heat equation with boundary control and observation: I. Modeling as port-Hamiltonian system

Keywords: Port-Hamiltonian System, Heat Equation, Thermodynamics, Boundary Control

, extra physical variables are introduced allowing to recover the first principle. The third formulation is classical from a mathematical perspective, although less meaningful physically speaking; however the Hamiltonian proves to be a Lyapunov functional, which is useful for boundary control purposes. Moreover, all these three formulations can be discretized with a structure-preserving scheme, as presented in the companion paper Serhani et al. (2019a).

INTRODUCTION

Port-Hamiltonian systems (pHs) have proved very efficient for the modeling and control of complex multiphysics systems [START_REF] Maschke | Port-controlled Hamiltonian systems: Modelling origins and system theoretic properties[END_REF]; [START_REF] Duindam | Modeling and Control of Complex Physical Systems: The Port-Hamiltonian Approach[END_REF]). It allows in particular to tackle the problem of model-based controllers design. The approach relies on a functional, namely the Hamiltonian, which encodes physical information about the system; a geometrical interconnection stucture is another key ingredient of the approach. This leads to a useful power balance, and in particular to the relevant ways to control and observe the system (w.r.t. this choice of Hamiltonian). Modularity is a salient feature of pHs: interconnecting two of them through their ports (boundary input/control and boundary output/observation for instance) leads to a new pHs. A Hamiltonian for the obtained pHs is easily computed from the Hamiltonians of the subsystems. Hence the power balance of complex systems can be obtained from the power balances of the simpler subsystems. Furthermore, the pHs formalism allows to tackle non-linear systems (van der [START_REF] Van Der Schaft | Port-Hamiltonian systems theory: an introductory overview[END_REF]).

It has to be noted that this formalism lies half-way between physics and mathematics. It aims at taking into account the geometrical structure of the system under consideration, together with the physical meaning of the variables or fields used for its description. In particular, it put aparts physical laws (such as conservation law, balance equation, etc.), parameters (such as diffusivity coefficient, mass density, etc.), and constitutive relations (such as Ohm's law, Fourier's law, etc.).

The terminology pHs comes from dynamical systems driven by Hamilton's equation in finite dimension. The energy variables are the generalized momenta and positions, and the flows are their time derivatives. Taking the mechanical energy as Hamiltonian, one can derive the efforts through its gradient and obtain a system of the form (1) below: see e.g. [START_REF] Van Der Schaft | Port-Hamiltonian systems theory: an introductory overview[END_REF] for an introduction to this formalism.

In the last two decades, a wide literature has grown around pHs formalism. It has been generalized to infinitedimensional systems (van der Schaft and [START_REF] Van Der Schaft | Hamiltonian formulation of distributed-parameter systems with boundary energy flow[END_REF]; [START_REF] Duindam | Modeling and Control of Complex Physical Systems: The Port-Hamiltonian Approach[END_REF]; [START_REF] Jacob | Linear port-Hamiltonian systems on infinite-dimensional spaces[END_REF]), to tackle space-time dynamics. Other types of Hamiltonians than the physical energy have been used (see for instance van der Schaft and Maschke ( 2018)), leading to different kinds of information about the system under study. We can note that the description we use here is not the more general framework of pHs, but it is sufficient for our purpose. A more recent topic is to provide accurate discretization methods (both in time and space) to preserve this powerful formalism. Several strategies have been tested to this aim, as the finite difference method [START_REF] Kotyczka | Structured discretization of the heat equation: Numerical properties and preservation of flatness[END_REF]), the finite volume method [START_REF] Kotyczka | Weak form of Stokes-Dirac structures and geometric discretization of port-Hamiltonian systems[END_REF]) and the finite element method (Cardoso-Ribeiro et al. ( 2018)) for instance. The latter is both an easy to implement and accurate way to proceed in so far as it can be structure-preserving when making use of the Partitioned Finite Element Method (PFEM), as will be proved in the companion paper Serhani et al. (2019a); this method is fully detailed in [START_REF] Cardoso-Ribeiro | A partitioned finite-element method (PFEM) for powerpreserving discretization of open systems of conservation laws[END_REF] for conservative systems, and has recently been extended to dissipative systems in Serhani et al. (2019c).

In this work, we aim to apply the pHs formalism to the heat equation in the anisotropic and heteregenous case, in an n-D geometrical domain; the choice of three different Hamiltonian functionals leads to three different points of view, which prove complementary. The first two are thermodynamically founded: the entropy and the internal energy, while the third one is less meaningful physically speaking, but enjoys an interesting behavior on a mathematical level. The latter should be rather considered as a candidate Lyapunov functional for boundary control purposes. In the three cases, thermodynamical laws (i.e. constitutive relations) are needed to close the system; they strongly rely on the context (we can think for instance of Dulong-Petit, Einstein or Debye models, linking the internal energy to the temperature, and to Fourier's law or Cattaneo's law linking the heat flux to the temperature). In other words, relevant constitutive relations are needed for the physical case under study. However, with the pHs formalism, since constitutive relations are put apart, this need can happily be postponed. As will be seen in Serhani et al. (2019a), an interesting consequence at the numerical level is that the work on the discretization of the physical laws comes first, while the discretization of the constituve relations comes at the end of the strategy.

Mathematically, the Hamiltonian H depends on what are called energy variables, typically extensive variables for the problem under study. Taking the variational derivatives (an analogue of the gradient in finite dimension) of H with respect to the energy variables gives the co-energy variables, typically intensive variables. Using physical laws, these variables allow to write a system into the following form (which can be called boundary pHs)

- → f = J - → e , v ∂ = B - → e , y ∂ = C - → e , (1)
where

• - → f are the flows, • - → e are the efforts, • J is a formally skew-symmetric operator, • B is a control operator, • C is an observation operator, • v ∂ is the control (or input, or actuation),
• y ∂ is the observation (or output, or measurement).

All vector fields and related quantities will be denoted in bold with an arrow.

Parameters are contained in the definitions of -→ f and -→ e , while J is a structure operator which can involve spatial derivatives but does not contain any parameter. Moreover, some extra constitutive relations between -→ f and -→ e are needed to close the system (possibly introducing new physical parameters) in order to be able to solve it.

The operators B and C are typically boundary operators, such as boundary trace operators (hence the name boundary pHs and the ∂ index for v ∂ and y ∂ ). In suitable functional spaces, B and C have to satisfy a relation with respect to the operator J ; roughly speaking this relation corresponds to the default of skew-adjointness. We do not enter in these mathematical considerations further, since this is not the topic of this paper (see for instance [START_REF] Kurula | Linear wave systems on n-D spatial domains[END_REF]). Keep in mind that this is closely related to the following Green formula in our case of interest:

Ω f div( - → g ) + Ω ---→ grad(f ) • - → g = ∂Ω f - → g • - → n ,
where Ω ⊂ R n is an open connected set and -→ n its outward unit normal.

The paper is organized as follows. In Section 2, the thermodynamical setting is presented and the useful notations are introduced. Then, the three continuous models, based on physical laws, are rewritten in the pHs formalism. In Section 3, the choice of entropy as Hamiltonian leads to an anti-dissipative closed system, or an accrescent open system. In Section 4, the choice of internal energy as Hamiltonian leads to a conservative closed system, or a lossless open system. In Section 5, the choice of a weighted L 2 functional of the internal energy density leads to a dissipative closed system, or a lossy open system. We conclude in Section 6 with a discussion on the comparison of the three points of view, and propose some interesting perspectives.

THERMODYNAMICAL HYPOTHESES AND NOTATIONS

All the thermodynamical material listed in this section can be found in e.g. [START_REF] Bird | Transport phenomena[END_REF].

Let Ω ⊂ R n be a bounded open connected set, n ≥ 1, with mass density ρ(x), for all x ∈ Ω; and -→ n denotes the outward unit normal on the boundary ∂Ω. We suppose that this domain does not change over time, i.e. we work at constant volume. No chemical reactions are to be found in the domain.

Let us denote

• u the internal energy density,

• - → J Q the heat flux, • T the local temperature, • β := 1 T the reciprocal temperature, • s the entropy density, • - → J S := β - → J Q the entropy flux, • C V := du dT V the isochoric heat capacity.
The first law of thermodynamics reads

ρ(x)∂ t u(t, x) = -div - → J Q (t, x) , ∀t ≥ 0, x ∈ Ω. (2)
Under our hypothesis, Gibbs formula reads dU = T dS, i.e.

∂ t u(t, x) = T (t, x) ∂ t s(t, x), ∀t ≥ 0, x ∈ Ω. (3)
Considering u as a function of the entropy s, and using Gibbs formula (3), we get for all t ≥ 0 and all

x ∈ Ω ρ(x) ∂ t s(t, x) = -div - → J S (t, x) + σ(t, x) , (4)
where

σ := ---→ grad (β) • - → J Q is the irreversible entropy production.
In the sequel, we will often take a look at what is obtained when using Fourier's law as constitutive relation:

- → J Q (t, x) = -λ(x) • ---→ grad (T (t, x)) , ∀t ≥ 0, x ∈ Ω, (5)
where λ is a tensor representing the thermal conductivity; it is a positive symmetric tensor thanks to Onsager's reciprocal relations. Note that the use of a tensor enables dealing with anisotropic media. Usual (i.e. isotropic) Fourier's law can be recovered with λ = λ 0 I, where I is the identity tensor.

With the above material, and in particular assuming (5), the classical PDE for the temperature T can be easily recovered: for all t ≥ 0 and all x ∈ Ω,

ρ(x)C V (t, x) ∂ t T (t, x) = div λ(x) • ---→ grad (T (t, x)) . (6)
This equation will serve later as a reference for the numerical simulations, since it is classical and easy to discretize with the finite element method (at least when C V is timeinvariant, as supposed in the Dulong-Petit model).

ENTROPY FORMULATION

Let us see the entropy density s as a function of the internal energy density u, and define the entropy of the system as Hamiltonian functional, i.e.

S(t)

:= Ω ρ(x) s(u(t, x)) dx , (7)
for all t ≥ 0, with u as energy variable. By definition, the co-energy variable is given by the variational derivative of S w.r.t. u in the weighted L 2 ρ (Ω) space (i.e. with measure ρ(x) dx, an important point to take in account when making use of Riesz representation theorem); by Gibbs formula (3), we get:

δ u S = ds du = β .
Thus, thanks to (3) and (4), one computes

d t S(t) = Ω ρ(x)∂ t u(t, x)β(t, x) dx, = Ω ρ(x)∂ t s(t, x) dx, = - Ω div - → J S (t, x) dx + Ω σ(t, x) dx, = - ∂Ω - → J S (t, γ) • - → n (γ) dγ + Ω σ(t, x) dx, = - ∂Ω β(t, γ) - → J Q (t, γ) • - → n (γ) dγ + Ω σ(t, x) dx .
The last equality comes from the definition of -→ J S . This relation expresses the power balance associated to the entropy as Hamiltonian. In particular, by the second law of thermodynamics, we can see that Ω σ has to be positive. However, this requires a constitutive relation, for instance between ---→ grad(β) and -→ J Q (in view of the definition of σ), to conclude. With Fourier's law (5) as constitutive relation, we obtain

d t S(t) = ∂Ω β(t, γ) λ(γ) • ---→ grad (T (t, γ)) • - → n (γ) dγ + Ω β(t, x) 2 ---→ grad (T (t, x)) • λ(x) • ---→ grad (T (t, x)) dx.
So that Ω σ is indeed positive, since λ is a positive symmetric tensor.

Moreover, the latter power balance suggests one possible definition for the boundary ports of the pHs in terms of the Dirichlet trace of the reciprocal temperature β, and the normal trace of the heat flux -→ J Q at the boundary ∂Ω of the domain.

Define f u := ∂ t u as the first flow, associated to the energy variable u. By definition of the co-energy variable, the dual effort is e u := β. Since the relation ρf u = -div -→ J Q must be fulfilled, we also define the effort -→ e Q := -→ J Q . Thus, with the following definition

- → f Q := - ---→ grad (β), this leads to ρf u - → f Q = 0 -div - ---→ grad 0 e u - → e Q . ( 8
)
Remark that no additional hypothesis is needed at this stage. Flows and efforts are defined in order to make a link between the quantities through a skew-symmetric operator. In particular, the equality

- → f Q = - ---→
grad(e u ) (which in turn is only algebraic and not dynamic) will not help solve the system without resorting to an extra constitutive relation linking -→ f Q and -→ e Q (such as Fourier's law (5) for instance). At first glance, this could seem a bit redundant, even unnecessarily complicated, but it leads to the powerful structure (1) described in the introduction.

A technical point to discuss is why f u is defined by f u := ∂ t u, and not by f u := ρ∂ t u? Indeed, the latter definition seems more straightforward, in view of (1). However, we work in L 2 ρ to define e u . Thus f u := ∂ t u indeed, but in L 2 ρ (Ω), roughly speaking, and so (1) is true in appropriate functional spaces. The multiplication of the flow f u by ρ is then done, so as to work back in the L 2 (Ω) space, which will be more practical in the discretization process.

Concerning the choice of boundary ports, we refer to Le [START_REF] Gorrec | Dirac structures and boundary control systems associated with skew-symmetric differential operators[END_REF] for a full caracterization of all the possible boundary ports for 1D linear systems. Here in n-D, there are at least two straightforward choices, although more general ones could be investigated. For the first choice, B -→ e := e u| ∂Ω , and With these notations at hand, we get the power balance in terms of the pHs variables

C - → e := -( - → e Q • - → n ) | ∂Ω , i.e.
d t S(t) = ∂Ω v ∂ (t, γ) y ∂ (t, γ) dγ - Ω - → f Q (t, x) • - → e Q (t, x) dx. (9) With -Ω - → f Q (t, x) • - → e Q (t, x) dx = Ω σ ≥ 0,
(t)) L 2 (∂Ω) = -∂Ω e u| ∂Ω ( - → e Q • - → n ) | ∂Ω dγ.

ENERGY FORMULATION

Now, let us see the internal energy density u as a function of the entropy density s, and define the internal energy of the system as Hamiltonian functional, i.e.

U (t) := Ω ρ(x) u(s(t, x)) dx , (10)
for all t ≥ 0, with s as energy variable. The same strategy as in the previous section is now being applied.

The co-energy variable is given by the variational derivative of U w.r.t. s:

δ s U = du ds = T .
Hence, from Gibbs formula (3) and ( 2), one computes

d t U (t) = Ω ρ(x)∂ t s(t, x)T (t, x) dx, = Ω ρ(x)∂ t u(t, x) dx, = - Ω div - → J Q (t, x) dx, = - ∂Ω - → J Q (t, γ) • - → n (γ) dγ, = - ∂Ω T (t, γ) - → J S (t, γ) • - → n (γ) dγ.
This latter power balance suggests to define the boundary ports of the pHs in terms of the Dirichlet trace of the temperature T , and the normal trace of the entropy flux -→ J S at the boundary ∂Ω of the domain.

Define f s := ∂ t s as the first flow, associated to the energy variable s. By definition of the co-energy variable, the dual effort is e s := T . Since ρf s = -div -→ J S + σ must be fulfilled, we also define the effort -→ e S := -

→ J S . Now, defining - → f S := - ---→ grad (T ) leads to: ρf s - → f S = 0 -div - ---→ grad 0 e s - → e S + σ 0 .
In this formulation though, σ appears as a right-hand side, and seems to be disconnected from the physical system itself, whereas it should be intimately linked to it, since [START_REF] Zhou | Distributed port-Hamiltonian modelling for irreversible processes[END_REF]; [START_REF] Zhou | Approche thermodynamique pour la commande d'un système non linéaire de dimension infinie : application aux réacteurs tubulaires[END_REF], the system can be extended by introducing a new effort variable e σ := -σ and its related flow f σ := T , in order to obtain

σ := ---→ grad (β) • - → J Q . Hence, following
  ρf s - → f S f σ   =   0 -div -1 - ---→ grad 0 0 1 0 0     e s - → e S e σ   . (11)
Note again that no additional hypothesis is needed at this stage, but only the definition of flows and efforts fitting into the pHs framework. Obviously, the J operator obtained in ( 11) is formally skew-symmetric.

Concerning the possible definition of the boundary ports, here again there are at least two choices, even though more The proof is quite lengthy, but it closely follows (Zhou et al., 2017, Prop. 4.2), so it is not being reproduced here.

Proposition 1 implies that

Ω ρf s e s + - → f S • - → e S + f σ e σ dx = ∂Ω v ∂ y ∂ dγ .
System (11) has one dynamical equation, and two algebraic equations. In order to solve them, one initial condition and two closure equations are needed. Indeed, the definition of σ translates into the first closure equation:

- → f S • - → e S + f σ e σ = 0 . (12)
Moreover, Fourier's law gives a second closure equation:

e s - → e S = λ • - → f S .
For a thorough thermodynamically oriented interpretation of these two closure equations, see the enlightening remark (Zhou et al., 2017, Rem. 1), which does also apply to our context.

With these notations at hand, we get the following power balance in terms of the pHs variables:

d t U (t) = ∂Ω v ∂ (t, γ) y ∂ (t, γ) dγ. ( 13
)
Note that, according to the first law of thermodynamics, the internal energy of the isolated system is constant, the closed system is conservative, i.e. d t U (t) = 0, while the open system is lossless, i.e.

d t U (t) = (v ∂ (t), y ∂ (t)) L 2 (∂Ω) = -∂Ω e s| ∂Ω ( - → e S • - → n ) | ∂Ω dγ.

LYAPUNOV FORMULATION

This last section is certainly less accurate from a thermodynamical point of view. But still, it does give some interesting features, widely used and known in the applied mathematics community. It has already been used, e.g. in [START_REF] Macchelli | Port Hamiltonian formulation of infinite dimensional systems I[END_REF], for the heat equation in the pHs formalism, though in the restrictive case of constant coefficients and 1D geometry. The more general n-D case with space varying coefficients is presented here, allowing to recover easily (6). This third formulation deserves some interest in its own right, and will be most useful at the numerical stage also.

Let H(t) := 1 2 Ω ρ(x) (u(t,x)) 2 C V (t,x) dx be the Hamiltonian and u the energy variable. The co-energy variable is δ u H = u C V . One easily computes from (2)

d t H(t) = Ω ρ(x)∂ t u(t, x) u(t, x) C V (t, x) dx, - 1 2 Ω ρ(x)∂ t C V (t, x) (u(t, x)) 2 (C V (t, x)) 2 dx, = - Ω div - → J Q (t, x) u(t, x) C V (t, x) dx, - 1 2 Ω ρ(x)∂ t C V (t, x) (u(t, x)) 2 (C V (t, x)) 2 dx, = Ω - → J Q (t, x) • ---→ grad u(t, x) C V (t, x) dx - ∂Ω u(t, γ) C V (t, γ) - → J Q (t, γ) • - → n (γ) dγ - 1 2 Ω ρ(x)∂ t C V (t, x) (u(t, x)) 2 (C V (t, x)) 2 dx. Now let us define f u := ∂ t u, e u := u C V and - → e Q := - → J Q .
Then, using the same strategy as for the first two formulations, we also define

- → f Q := - ---→ grad u C V .
As in the entropy formulation, we get:

ρf u - → f Q = 0 -div - ---→ grad 0 e u - → e Q . ( 14
)
For the boundary ports, either B -→ e := e u| ∂Ω and C -→

e := -( - → e Q • - → n ) | ∂Ω , or B - → e := -( - → e Q • - → n ) | ∂Ω and C - → e := e u| ∂Ω .
Hence, the power balance is

d t H(t) = - Ω - → e Q (t, x) • - → f Q (t, x) dx + ∂Ω v ∂ (t, γ) y ∂ (t, γ) dγ - 1 2 Ω ρ(x)∂ t C V (t, x)(e u (t, x)) 2 dx. ( 15
)
At this stage however, to the best of our knowledge, we do not know any dynamic for C V , and the boundary value of u C V is of no use in practice.

With Dulong-Petit model, i.e. assuming that u(t, x) = C V (x) T (t, x), one immediately gets from (15)

d t H(t) = Ω - → J Q (t, x) • ---→ grad (T (t, x)) dx - ∂Ω T (t, γ) - → J Q (t, γ) • - → n (γ) dγ.
Here, the control and the observation can be chosen to be the boundary temperature and the boundary flux (or the other way), giving useful physical meanings to both these boundary ports for control purpose.

Finally, adding Fourier's law (5) as one needed constitutive relation, the power balance becomes in terms of the pHs variables

d t H(t) = - Ω - → f Q (t, x) • λ(x) • - → f Q (t, x) dx + ∂Ω v ∂ (t, γ) y ∂ (t, γ) dγ . ( 16
)
Since λ is a positive symmetric tensor, the closed system is dissipative, i.e. d t H(t) ≤ 0, while the open system is lossy, i.e.

d t H(t) ≤ (v ∂ (t), y ∂ (t)) L 2 (∂Ω) = -∂Ω e u| ∂Ω ( - → e Q • - → n ) | ∂Ω dγ.

CONCLUSION AND PERSPECTIVES

We saw three ways to describe the heat equation through the pHs formalism. We intend now to draw the pros and cons about these formulations.

This first important point to note is that the Lyapunov approach is theoretic, and not thermodynamically founded since the physical units of this Hamiltonian are J • K, which is not meaningful. Furthermore, it requires a demanding and restrictive constitutive relation -the Dulong-Petit model -to achieve its main purpose: obtaining the temperature and the heat flux as boundary control and observation. But still, this kind of boundary control and observation seems to be more accurate in practice, and this justifies to some extent the presentation of this formulation in this work.

The other two choices of Hamiltonians, thermodynamically founded, lead to a practical drawback: one of the boundary port seems, to the best of our knowledge, more difficult to realize in the experiments. More precisely, we do not know if there exist actuators and sensors to deal with the boundary reciprocal temperature or the boundary entropy flux.

Finally, we would point out that with the energy formulation, since we introduced a new flux/effort couple related to the irreversible entropy production σ, one could think of the need for an additional constitutive relation to solve the system. However, this closure equation is directly obtained by definition of σ, which leads to (12). Consequently, this leads to two algebraic equations for the final system. This is indeed another difficulty, but still this formulation has its own advantages: the Hamiltonian is an energy, and we compute directly one more variable, namely σ, not reachable without extra efforts in the other two formulations.

The immediate perspective of this theoretical work is to present a structure-preserving numerical method adapted to the three formulations of the heat equation, which is the main topic of the companion paper Serhani et al. (2019a).

  the boundary control is the reciprocal temperature, and the boundary observation is the heat flux. For the second choice, B -→ e := -( -→ e Q • -→ n ) | ∂Ω , and C -→ e := e u| ∂Ω , i.e. the boundary control is the heat flux, and the boundary observation is the reciprocal temperature. Note that the choice of the signs is a convention (control or observe the inward flux).

  general ones could be investigated. Either B -→ e := e s| ∂Ω , and C -→ e := -( -→ e S • -→ n ) | ∂Ω , i.e. we control the temperature and observe the entropy flux. Or B -→ e := -( -→ e S • -→ n ) | ∂Ω , and C -→ e := e s| ∂Ω , i.e. we control the entropy flux and observe the temperature. Proposition 1. Extended system (11) with the boundary ports v ∂ = B -→ e and y ∂ = C -→ e in the above definitions gives rise to a Stokes-Dirac structure.

  the closed system is anti-dissipative, i.e. d t S(t) ≥ 0, and the open system is accrescent, i.e. d t S(t) ≥ (v ∂ (t), y ∂

ACKNOWLEDGEMENTS

We thank Pr. F. Couenne from LAGEP for some helpful conversation and advice.

This work has been performed in the frame of the Collaborative Research DFG and ANR project INFIDHEM n • ANR-16-CE92-0028 (http://websites.isae.fr/infidhem).


R Pethe

T Heuze

L Stainier
email: laurent.stainier@ec-nantes.fr

Variational Approach for thermo-mechanical mesh adaptation

Keywords: variational, mesh adaptation, thermo-mechanical

Introduction

In many engineering problems (thermal or structural), zones of high gradient of fields of interest evolve with time and loading. When we solve such a problem with finite elements method, mesh has to be dynamically adapted in order to capture solution in zones of high gradients with required precision.

Many methods of mesh adaptation are proposed in the literature based on error estimators. But, these methods have limitations when we wish to treat problems with non-linear constitutive relations (e.g. plasticity).

An alternative approach of mesh adaptation was recently proposed [START_REF] Mosler | Variational h-adaptation in finite deformation elasticity and plasticity[END_REF] [2] based on variational approach [START_REF] Ortiz | Variational formulation of viscoplastic constitutive updates[END_REF]. This technique allows to perform calculations in the presence of large transformations and nonlinear irreversible behavior (e.g. plasticity).

In this work, we are focusing on steady state thermal problems. In this case, we try to minimize energy like potential Φ. Firstly, we calculate the finite element solution on initial mesh. Secondly, on each element, we calculate the local value of energy like potential. Thirdly, we add a node in middle of the element. fourthly, we solve local problem on the element by fixing the temperature at end nodes. (Therefore, it becomes a 1D problem.) Then, we check if we improve significantly the energy like potential locally by adding this node. If we do, we add this node in the global mesh. Finally, we solve whole problem again on the new mesh obtained. We repeat the procedure till we get a solution of required precision.

Variational Formulation

The solution for steady thermal problem is found my minimizing the following convex potential:

φ(T (x)) = Ω 1 2 k --→ gradT • --→ gradT dΩ - Ω rT dΩ ( 1
)
where k is the thermal conductivity, r the internal heat source, T the temperature field and Ω the domain. If we are dealing in 1D, the strong form reads:

k d 2 T dx 2 + r = 0 (2)
with boundary conditions:

T (x = 0) = T 0 T (x = L) = T 1 (3)

Applications

Now, we try to apply this theory to thermal stationary problem. (The transient problem will be presented during the conference.)

Analytical Solution

The equation to be solved in stationary problem can be represented as follows:

k d 2 T dx 2 + r = 0 (4)
with boundary conditions:

T (x = 0) = T 0 T (x = L) = T 1 (5)
Here, k is the thermal conductivity of the material, T is the temperature field, r is the internal heat generation, L is the length of the rod, and q is a constant. We take,

r = x q T 0 = 0 T 1 = 0 (6)
Therefore, the analytical solution of the problem can be given as follows:

T = L q+1 x (q + 1)(q + 2) - x q+2 (q + 1)(q + 2) (7)
The energy potential is given by,

U = k 2 L 2q+3 ( 1 (q + 1) 2 (2q + 3) - 1 (q + 1) 2 (q + 2) 2 )-L 2q+3 ( 1 (q + 1)(q + 2) 2 - 1 (q + 1)(q + 2)(2q + 3) ) (8)
By taking q = 51, we can represent the analytical solution graphically as shown in figure 1. From figure 1, one can observe the sharp gradient of the field on the extreme right part of the bar.

Numerical Solution

We solved the problem numerically by taking q = 51. The solution can be represented as shown in figure 2. From figure 2, one can observe that on the right hand side, where we have sharp gradient, we have greater number of elements to capture solution better. However, in the remaining part, algorithm puts fewer elements which are sufficient to represent the solution.

Analysis

In order to assess the usefulness of this algorithm, we plot the error in calculated solution with respect to the number of nodes in log-log scale. We consider 3 cases. In the first case, we plot the error at each refinement iteration in adaptive mesh algorithm with respect to the cumulative number of nodes. We accumulate the number of nodes in this case because it will give us idea of not only the cost of calculating result on a particular mesh, but also it adds the cost of reaching to this particular mesh. In the second case, we plot error at each refinement iteration in adptative mesh algorithm with respect to the number of nodes in the mesh and in the third case, we plot error in uniform mesh refinement with respect to the number of nodes of each case. Figure 3 shows this plot for L2 norm of error in temperature field and figure 4 shows plot for error in energy norm error in the energy like potential.

Conclusions

In this work, we explained the variational approach for mesh refinement. We formulated the problem. Then we applied our hypothesis on a 1-D thermal problem.

In figures 3 and 4, we plotted L2 error norm in temperature field vs. number of nodes and energy error norm in energy vs. number of nodes respectively. In these graphs, we have also plotted the error vs. uniform refinements. The idea is that when mesh adaptation curve lies below the curve of uniform refinement, we have a better algorithm.(Less cost with equal or better precision.) Therefore, we plotted one curve with number of nodes accumulated and another with number of nodes. If we consider energy error as in figure 4, we see that variational algorithm is always better. However, if we consider error in temperature 3, the curve with non-accumulated nodes lies below the curve of refinement. But, the curve in which we accumulate the number of nodes, is first above the curve of uniform refinement, and as the algorithm progresses, it goes below. This behavior is logical because, the regions where variational adaptation curve is above the curve of uniform refinement, the mesh is too coarse.

Finally, we show that the variational mesh adaptation algorithm costs us less than a uniform mesh and still gives us better results.

The next part in the thesis is to apply this technique to the transient thermal problem and coupled thermo-mechanical problem using thermo-elasticity(Results of this part

Figure 1 :

 1 Figure 1: representation of analytical solution.

Figure 2 :

 2 Figure 2: representation of numerical solution.

Figure 3 :

 3 Figure 3: L2 norm error in temperature field with respect to number of nodes.

Figure 4 :

 4 Figure 4: energy norm error in energy like potential with respect to number of nodes.

will be shown in conference). Finally, we should extend this approach in 2D and 3D.


Jonathan Bell
email: belljonathan50@gmail.com

Networked Head-Mounted Displays for Animated Notation and Audio-Scores with SmartVox

Keywords: Networked Music Performance, Computer-aided performance, Computer-aided composition, Musical notation, Head-Mounted Display, Augmented Reality CCS Concepts, Networks → Local area networks, •Human-centered computing → Smartphones, •Applied computing Computer-assisted instruction, Performing arts, Sound and music computing

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION: WHAT IS SMARTVOX?

SmartVox [4][3] is a distributed web application that delivers and synchronizes audiovisual scores in the mp4 format to the performer's mobile devices, in compositions involving up to 80 simultaneous performers such as in Le temps des Nuages. Recent developments include the use of headmounted displays (HMDs) for technology-aided performance, as in the pieces In Memoriam Jean-Claude Risset I (Figure 1) and Mit Allen Augen, In Memoriam J.C. Risset II (Figure 2).

Technical implementation

SmartVox was developed in the SoundWorks framework. [START_REF] Bell | Audio-scores, a resource for composition and computer-aided performance[END_REF] SoundWorks provides a set of services -such as synchronization, network messages, distributed states, creation of groups of clients -that aims to solve problems common to distributed and synchronized web applications centered on multimedia rendering. The framework is written in Javascript, with a server side based on Node.js. [START_REF] Bell | Audivisual Scores and Parts Synchronised over the Web[END_REF] The SmartVox application consists of two web clients, the player and the conductor, that can be executed in any recent web browser on mobile devices (e.g. smartphones, tablets) and laptops. The realtime communication between clients is achieved through the WebSocket protocol [START_REF] Bell | SmartVox. A web-based distributed media player as notation tool for choral practices[END_REF] The application is typically deployed [START_REF] Agostini | Bach: an environment for computer-aided composition in max[END_REF] Those three pieces are respectively available at: https://youtu.be/SyFdR2HiF00, https://youtu.be/hQtyu1dcCaI, and https://youtu.be/ET OBgFWx04. [START_REF] Bell | Audio-scores, a resource for composition and computer-aided performance[END_REF] SoundWorks was initiated by the CoSiMa research project funded by the French National Research Agency (ANR) and coordinated by Ircam. [START_REF] Bell | Audivisual Scores and Parts Synchronised over the Web[END_REF] https://nodejs.org/en 4 https://www.w3.org/TR/WebSockets/ over a local area network, but it may also be used over the internet. [START_REF] Carey | From Mimetics to Memetics[END_REF]

Composer's Score Elaboration Workflow

The audiovisual scores (mp4 files) used in Smartvox are composed in the Bach environment [1], a Max library for real-time computer-aided composition. In this environment (bach.roll or bach.score), each note of the score can store metadata in its slot content (e.g. Path to a sample, automation curve...). This feature is used to control synthesizer that creates the audio cues, directly from the notational environment [START_REF] Conan | An intuitive synthesizer of continuous-interaction sounds: Rubbing, scratching, and rolling[END_REF] . Each separate part (one part per voice/instrument) of audiovisual notational material generated in Bach is then recorded by screen/audio captures, in order to be distributed and synchronized by SmartVox in rehearsals and performance, as shown in the diagram below:

Performer's Perception

From a singer's or instrumentalist's point of view, SmartVox provides simultaneously auditory (audio-score [2]) and visual (animated notation) information, which facilitates the interpretive and performative processes by recalling of the performers' separate parts, thus expediting the learning and rehearsal process of a new work. This synergy between visual and acoustic stimuli also enables spatial separation or movement of the performers, and allows for the exploration of unfamiliar musical materials (e.g. microtonal tuning, texts in a foreign language).

SMARTVOX FOR MIMETIC ORCHES-TRATION OF ELECTRONICS 2.1 Frequency Domain : Spectral Harmony

SmartVox was initially imagined as a tool that would help choristers to tune with the electronics. In spite of its initially loose synchronization capabilities, it quickly appeared that, with singers of all levels, the system allowed for a precise control of harmony in the context of choral music in particular. [START_REF] Fober | Distributing music scores to mobile platforms and to the internet using inscore[END_REF] The piece Smartvox8 (which has the same name as the application) remains the best illustration of it capabilities in spectral terms.

Time Domain: Rhythmic Precision

At IRCAM in January 2018, Benjamin Matuszewski improved the synchronization possibilities of SmartVox by implementing a client-side algorithm that puts back in sync devices whose drift exceeds a certain threshold ([3], Chapter 4.3). Thanks to this new implementation, it can now explore a wide range of tightly synchronized musical situations. These new possibilities encouraged the author to write rhythmical games between performers, taking advantage of the cursor-type display of time, as in In Memoriam Jean-Claude Risset 1 , 9 or sharper articulations, such as the clapping game present throughout the piece Mit Allen Augen, in memoriam J.C. Risset II . [START_REF] Gabrielli | Wireless Networked Music Performance[END_REF] During the composition process, 11 the precise lining-up and synchronization of live performers with electronics was also improved by ability to use Bach as a master over Ableton : the bach.roll object displays notation in proportional time, and outputs notifications of its playback status in real-time. These notifications can be interpreted in Max For Live in order to synchronize 1/ the notation for human players in Bach and 2/ the electronics in Ableton Live 12 .

Conclusion : Mimetic Orchestration

A great majority of the electronic material presented in the Risset cycle [START_REF] Hope | The decibel scoreplayer -a digital tool for reading graphic notation[END_REF] constitutes mimetic 14 orchestrations of the PRISM laboratory synthesizer, developed by Richard Kronland-Martinet [6]. The recent improvement of SmartVox in terms of synchronisation now allow more flexibility for musical expermentation of various sorts. In Das Hoheslied, In Memoriam Jean-Claude Risset 3 for instance, students of HfMT Hamburg were able to sight-sing (for the very first time) passages which strikingly resembled the target model. [START_REF] Nathan | Musician assistance and score distribution (masd)[END_REF]

DISTRIBUTED NOTATION 3.1 Distributed Notation: Global Context

Composers and researchers increasingly acknowledge the strong analogy which can be drawn between the traditional 'score and parts' musical practice led by a conductor, and the modern distributed systems or web applications (MASD [15] -SmartVox [4]), in which multiple clients coordinate their actions by passing messages to one another. Consequently many performance-oriented systems today (IN-Score [7], SmartVox [4][3], Decibel [14] [13], Maxscore [12]) endeavor to distribute and synchronize each part of the score on the performer's devices (whether Smartphones, tablets or laptops). The Tenor 2018 conference in Montreal revealed to the author an interesting similarity between the Decibel ScorePlayer and SmartVox, in the sense that the two softwares are rather elementary solutions both converging towards the score player performance-oriented paradigm, whilst other notation packages (Bach, Maxscore, Symbolist...) were also designed for more elaborate computer-aided composition and real-time processes. SmartVox reveals itself to be very effective for this kind of setup by rendering the score in the browser directly, without the need for installing a client application. Thanks to cross-platform web technologies, the application works with any browser capable [START_REF] Fober | Towards dynamic and animated music notation using inscore[END_REF] A rhythmical passage in In Memoriam Jean-Claude Risset 1 https://youtu.be/hQtyu1dcCaI?t=349 [START_REF] Gabrielli | Wireless Networked Music Performance[END_REF] See https://youtu.be/ET OBgFWx04?t=618. [START_REF] Hajdu | Embodiment and disembodiment in networked music performance[END_REF] The compositional stage is made in Bach and Ableton, as opposed to the performance during which only SmartVox is running [START_REF] Hajdu | Maxscore: Recent developments[END_REF] The following example shows the convenience of the Bach/Ableton inter-application communication https://youtu.be/VJvY5wYl cM [START_REF] Hope | The decibel scoreplayer -a digital tool for reading graphic notation[END_REF] In Memoriam J.C. Risset, Mit Allen Augen, In Memoriam J.C. Risset II , Das Hoheslied, In Memoriam Jean-Claude Risset III [START_REF] Hope | Scoring an animated notation opera -the decibel score player and the role of the digital copyist in 'speechless[END_REF] A term attributed to Lindsay Vickery [16] and Benedict Carey [5]. [START_REF] Nathan | Musician assistance and score distribution (masd)[END_REF] A Target sound is followed by its imitation by the choir in a sight-reading session of Das Hoheslied, In Memoriam Jean-Claude Risset 3. See https://youtu.be/EHYq9nFF6sE for demonstration. device, and no installation is required by the client. The node.js [START_REF] Vickery | Hybrid real/mimetic sound works[END_REF] /WebSocket architecture of SmartVox will hopefully inspire more composers, researchers and developers to investigate the emerging musical practice of distributed notation.

SmartVox.eu, Score Distribution over the Web and Remote NMPs

With the exponential growth of the web, hosting SmartVox17 on the internet (i.e. on a remote server rather than a local one) appeared as a necessity. However we demonstrated (in Chapter 4.2 'Measurements of timing accuracy' [3]) that, although the synchronization of different parts was quite accurate in this way, local solutions remained safer (in the same room, over Wi-Fi). Also, notwithstanding the fact that SmartVox belongs to the realm of Networked Music Performances, it became clear that its practical application falls under the 'local NMPs' sub-category defined by Gabrielli and Squartini [10]. Indeed in the case of SmartVox today, local NMPs still seem more viable, first technically (because of the unreliability of the internet in 2019), but also artistically speaking (because in a live performance the audience presumably expects the physical presence of the artist [11]).

The 'BabelBox', a 0-conf Raspberry Pi Solution for Local NMPs

The author has shown evidence that SmartVox is suitable for large scale projects such as le temps des nuages18 in which eighty singers and musicians had their score synchronized through the same network [3]. However, in the more intimate context of chamber music (as in In Memoriam JC Risset 1 ), the focus is slightly different: the network load is much lighter, which encouraged finding a minimal hardware solution, in order to make possible rehearsals without the physical presence of the composer or the mobilization of a technician only to setup a network. Installing the server on the performer's computer remotely has often been successful even with musicians unfamiliar with technology, thanks to the flexibility of the Node Package Manager (NPM), which reduces the installation of SmartVox to a few command lines. 19 This installation process nevertheless remained an obstacle for the dissemination of SmartVox. In search of a light plug-and-play dedicated system to be sent over the post, the Raspberry Pi quickly appeared as the best option to host SmartVox on an embedded system. node.js runs on Raspbian, and SmartVox proved to be very stable on a Raspberry Pi 3, so, once installed, the only two steps for a 0-conf deliverable hardware were:

• Setting up a static address for a dedicated router (e.g. tp-link...). • Starting SmartVox at boot. This low-cost system (less than 65 €, for a Raspberry and a router) now allows the sending of ready-to-use scores. Once the system is power-supplied, all the performers need to do is to join the dedicated Wi-Fi, and type the static IP address of the server on their smartphone/tablet (i.e. for the performers: 192.168.0.100:8000, and for the conductor: 192.168.0.100:8000/conductor). In January 2019, for the performance of a pedagogical piece, The Super Mario Babel Box . This small system was sent per post to the Caen French conservatoire via BabelScores 20 , thus proposing a rental of performing scores (separate parts) of a new kind. During the rehearsals and performances, synchronization prooved to be of great precision and download speed very fast, in spite of the lightness of the low-cost system.

BabelScores, Inscore and SmartVox

The representation of time in the age of screen scores is open to a wide array of solutions. In none-pulsed music particularly, and in spite of the great conducting tradition in chamber music and orchestral works, left-to-right scrolling cursors distributed on the performers' devices (as in the Decibel ScorePlayer or SmartVox) seem a far more straightforward strategy to obtain tight synchronization in comparison to the bars and beats 'encoding' (quantification by the composer with or without the help of algorithms) and decoding processes (a compromised interpretation by the instrumentalist, between the rhythmic values written on the page and the gestures of the conductor), inherited from a scoring tradition, in which a regular meter was assumed.

If Bach is the favoured environment for the author's own compositions, INScore [9][8] might be of particular relevance to extend the practice of distributed notation to a wider community of composers and performers, through the wide community of composers edited edited at BabelScores in particular. INScore allows for the precise temporal control of animated cursors in the graphical domain. Thanks to its OSC support, it is possible to control the cursor's position via automations in Ableton Live 21 . These animation can be used for scrolling cursors over standard notation 22 , and the arm movements of a conductor can be replaced by a bouncing ball 23 . These tools might be used, in combination with the BabelBox, for highly demanding pieces in terms of synchronization 24 , or as a tool for appreticeship of unfamiliar idioms such as early music notation. 25

HEAD MOUNTED DISPLAYS

In Memoriam JC Risset 1 26 , premiered in September 2018 at the Gaudeamus Festival (Utrecht), constitutes the author's first experiment using head-mounted displays. By simply displaying each part of the score over the heads of the performers (for flute and clarinet only), this piece revealed interesting potentials, further exploited on a larger scale in Mit Allen Augen, In Memoriam J.C. Risset II .

Just as traditional scores placed on a music stand, screenscores displayed on a tablet (for instrumentalists) or on a phone (for singers) oblige musicians to look and orientate their body constantly in the direction of the score. This well-established convention of the classical concert setup considerably limits the possibilities of staging music, in a theatrical context for instance.

According to the performers' feedback, head-mounted setups provide a large and comfortable display, since the 20 Babelscores (https://www.babelscores.com/) is an online score database for classical contemporary music, currently actively supporting the SmartVox project: environment is still visible around the score, or through the score in the case of holographic display (such as the Aryzon headset for instance, see Figure 4). The performers also showed interest in their ability to move freely on stage oraround the audience.

CONCLUSION

Initially conceived as a rehearsal tool for choral practices almost exclusively [4] [3], recent use cases have demonstrated that SmartVox is in fact better and more broadly described as a distributed score player suitable to instruments as well as vocalists with, or without electronics. If musical notation remains a key concept for SmartVox, the notion of Networked Musical Performance probably defines it more specifically. With its promising use of distributed head-mounted displays, SmartVox wishes to shift the focus from human-computer interaction towards computer-mediated human interaction and proposes an original interface for musical expression for collective technology-aided live performances.

1

 1

Figure 2 :

 2 Figure 2: Singer wearing HMD for technology-aided performance.

Figure 3 :

 3 Figure 3: Score elaboration workflow.

Figure 4 :

 4 Figure 4: Score display from the performer's point of view. While awaiting more discrete solutions (such as Vufine 27 Glasses), invisible from an audience's point of view, HDMs open vast perspectives for further research, which could expand even beyond the mere display of animated notation.

https://youtu.be/83ub6-Q5oj0

See https://youtu.be/s4qS2khwkT0 for demonstration.

See https://youtu.be/rlmqnIJhayU for demonstration.

See https://youtu.be/JZsJn7EEW-A.

Server-side JavaScript, see: https://nodejs.org/en/

Each instrumental/vocal part of the piece And the Sea is accessible through the following url www.smartvox.eu, and can be accessed simultaneously from e.g., an iPad for the flute, Android tablets for piano and cello, and a phone for the singer. A trailer of the premiere of the piece is accessible here: https://youtu.be/prcXUbhd-ZY

A recording of the piece is available at the following address: https://youtu.be/SyFdR2HiF00

SmartVox is open source and ready to download on GitHub: https://github.com/belljonathan50/SmartVox0.1.

ACKNOWLEDGMENTS

Andrea Agostini, Richard Baker, Laurence Brisset, Dominique Fober, Daniele Ghisi, Georg Hajdu, Richard Kronland-Martinet, Benjamin Matuszewski and Norbert Schnell.