Yeow Marc Plantevit

Wei Choong

Anne Laurent

Dominique Laurent

Maguelonne Teisseire

Y W Choong

M2SP: Mining Sequential Patterns Among Several Dimensions

Keywords: Data Mining, Sequential Patterns, Multidimensional Rules

à la diffusion 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

Mining sequential patterns aim at discovering correlations between events through time. For instance, rules that can be built are A customer who bought a TV and a DVD player at the same time later bought a recorder. Work dealing with this issue in the literature have proposed scalable methods and algorithms to mine such rules [START_REF] Srikant | Mining Sequential Patterns: Generalizations and Performance Improvements[END_REF]. As for association rules, the efficient discovery is based on the support which indicates to which extend data from the database contains the patterns.

However, these methods only consider one dimension to appear in the patterns, which is usually called the product dimension. This dimension may also represent web pages for web usage mining, but there is normally a single dimension. Although some works from various studies claim to combine several dimensions, we argue here that they do not provide a complete framework for multidimensional sequential pattern mining [START_REF] De Amo | An apriori-based approach for first-order temporal pattern mining[END_REF][START_REF] Pinto | Multi-dimensional sequential pattern mining[END_REF][START_REF] Yu | Mining sequential patterns from multidimensional sequence data[END_REF]. The way we consider multidimensionality is indeed generalized in the sense that patterns must contain several dimensions combined over time. For instance we aim at building rules like A customer who bought a sur f board and a bag in NY later bought a wetsuit in SF. This rule not only combines two dimensions (City and Product) but it also combines them over time (NY appears before SF, surfboard appears before wetsuit). As far as we know, no method has been proposed to mine such rules.

In this paper, we present existing methods and their limits from several studies in recent years. We define then the basic concepts associated to our proposition M 2 SP and the algorithms to build such rules. Experiments are performed on synthetic data to assess our proposition.

In our approach, sequential patterns are mined from a relational table, that can be seen as a fact table in a multidimensional database. This is why, contrary to the standard terminology of the relational model, the attributes over which a relational table is defined are called dimensions.

In order to mine such frequent sequences, we extend our approach so as to take into account partially instanciated tuples in sequences. More precisely, our algorithms are designed in order to mine frequent jokerized multidimensional sequences containing as few * as possible, i.e., replacing an occurrence of * with any value from the corresponding domain cannot give a frequent sequence.

This paper is organized as follows. Section 2 introduces a motivating example illustrating the goal of our work. Section 3 presents existing works concerning sequential patterns, multidimensional approaches and multidimensional databases. Section 4 introduces our contribution, M 2 SP, as an approach to discover multidimensional sequential patterns. Section 5 introduces jokerized patterns. Section 6 presents the algorithms. Section 7 presents results of experiments performed on synthetic data.

Motivating Example

In order to illustrate our approach, we consider the following example that will be used throughout the paper as a running example.

Let us consider a relational table T in which transactions issued by customers are stored. More precisely, we assume that T is defined over six dimensions (or attributes) as shown in Fig. 1, and where: D is the date of transactions (considering three dates, denoted by 1, 2 and 3), CG is the category of customers (considering two categories, denoted by Educ and Ret, standing for educational and retired customers, respectively), A is the age of customers (considering three discretized values, denoted by Y (young), M (middle) and O (old)), C is the city where transactions have been issued (considering three cities, denoted by NY (New York), LA (Los Angeles) and SF (San Francisco)), P is the product of the transactions (considering four products, denoted by c, m, p and r), Q stands for the quantity of products in the transactions (considering nine such quantities).

For instance, the first tuple of T (see Fig. 1) means that, at date 1, educational young customers bought 50 products c in New York. Let us now assume that we want to extract all multidimensional sequences that deal with the age of customers, the products they bought and the corresponding quantities, and that are frequent with respect to the groups of customers and the cities where transactions have been issued. To this end, we consider three sets of dimensions as follows: (i) the dimension D, representing the date, (ii) the three dimensions A, P and Q that we call analysis dimensions (tuples over analysis dimensions are those that appear in the items that constitute the sequential patterns to be mined), (iii) the two dimensions CG and C, that we call reference dimensions (the table is partitioned according to tuple values over reference dimensions and the support of a given multidimensional sequence is the ratio of the number of blocks supporting the sequence over the total number of blocks. Fig. 5 displays the corresponding blocks in our example).

In this framework, {(Y, c, 50), (M, p, 2)}, {M, r, 10)} is a multidimensional sequence having support 1 3 , since the partition according to the reference dimensions contains 3 blocks, among which one supports the sequence. This is so because (Y, c, 50) and (M, p, 2) both appear at the same date (namely date 1), and (M, r, 10) appears later on (namely at date 2) in the first block shown in Figure 4.

It is important to note that, in our approach, more general patterns, called jokerized sequences, can be mined. The reason for this generalization is that considering partially instanciated tuples in sequences implies that more frequent sequences are mined. To see this, considering a support threshold of 2 3 , no sequence of the form {(Y, c, µ)}, {(M, r, µ )} is frequent. On the other hand, in the first two blocks of Fig. 5, Y associated with c and M associated with r appear one after the other, according to the date of transactions. Thus, we consider that the jokerized sequence, denoted by {(Y, c, * )}, {(M, r, * )} , is frequent since its support is equal to 2 3 . 3 Related Work

D CG C A P Q (Date) (Customer-Group) (City) (Age) (Product) (

Sequential Patterns

An early example of research in the discovering of patterns from sequences of events can be found in [START_REF] Dietterich | Discovering patterns in sequences of events[END_REF]. In this work, the idea is the discovery of rules underlying the generation of a given sequence in order to predict a plausible sequence continuation. This idea is then extended to the discovery of finding interesting patterns (or rules) embedded in a database of sequences of sets of events (items). A more formal approach in solving the problem of mining sequential patterns is the AprioriAll algorithm as presented in [START_REF] Mannila | Discovering frequent episodes in sequences[END_REF]. Given a database of sequences, where each sequence is a list of transactions ordered by transaction time, and each transaction is a set of items, the goal is to discover all sequential patterns with a user-specified minimum support, where the support of a pattern is the number of data-sequences that contain the pattern. In [START_REF] Agrawal | Mining sequential patterns[END_REF], the authors introduce the problem of mining sequential patterns over large databases of customer transactions where each transaction consists of customer-id, transaction time, and the items bought in the transaction. Formally, given a set of sequences, where each sequence consists of a list of elements and each element consists of a set of items, and given a user-specified min support threshold, sequential pattern mining is to find all of the frequent subsequences, i.e., the subsequences whose occurrence frequency in the set of sequences is no less than min support. Sequential pattern mining discovers frequent patterns ordered by time. An example of this type of pattern is A customer who bought a new television 3 months ago, is likely to buy a DVD player now. Subsequently, many studies have introduced various methods in mining sequential patterns (mainly in time-related data) but almost all proposed methods are Apriori-like, i.e., based on the Apriori property which states the fact that any super-pattern of a nonfrequent pattern cannot be frequent. An example using this approach is the GSP algorithm [START_REF] Srikant | Mining Sequential Patterns: Generalizations and Performance Improvements[END_REF].

Data to be Mined

We assume that the reader is familiar with the relational model of databases [START_REF] Ullman | Principles of Database and Knowledge-Base Systems[END_REF] Since we are interested in sequential patterns, we assume that U contains at least one dimension whose domain is totally ordered, corresponding to the time dimension.

In our running example, we consider U = {D,CG, A, P, Q}, where D is the time dimension. Moreover, considering the table T of Fig. 1, we have for instance Dom(D) = {1, 2, 3} and Dom(CG) = {Educ, Ret}.

Multidimensional Sequential Patterns

As far as we know, three propositions have been studied in order to deal with several dimensions when building sequential patterns. Pinto et al. [START_REF] Pinto | Multi-dimensional sequential pattern mining[END_REF] is the first paper dealing with several dimensions in the framework of sequential patterns. For instance, purchases are not only described by considering the customer ID and the products, but also by considering the age, type of the customer (Cust-Grp) and the city where he lives, as shown in Fig. 1. Multidimensional sequential patterns are defined over the schema A 1 , ..., A m , S where the set of A i stands for the dimensions describing the data and S stands for the sequence of items purchased by the customers ordered over time. A multidimensional sequential pattern is defined as (id 1 ,(a 1 , ..., a m ),s) where a i ∈ A i ∪ { * }. id 1 ,(a 1 , ..., a m ) is said to be a multidimensional pattern. For instance, the authors consider the sequence (( * , NY, * ), b f ) meaning that customers from NY have all bought a product b and then a product f . Sequential patterns are mined from such multidimensional databases either (i) by mining all frequent sequential patterns over the product dimension and then regrouping them into multidimensional patterns, (ii) or by mining all frequent multidimensional patterns and then mining frequent product sequences over these patterns. Note that the sequences found by this approach do not contain several dimensions since the dimension time only concerns products. Dimension product is the only dimension that can be combined over time, meaning that it is not possible to have a rule indicating that when b is bought in Boston then c is bought in NY . Yu et Chen. [START_REF] Yu | Mining sequential patterns from multidimensional sequence data[END_REF] considers sequential pattern mining in the framework of Web Usage Mining. Even if they consider three dimensions (pages, sessions, days), these dimensions are very particular since they belong to a single hierarchized dimension. Moreover, the sequences found describe correlations between objects over time by considering only one dimension, which corresponds to the web pages. de Amo et al. [START_REF] De Amo | An apriori-based approach for first-order temporal pattern mining[END_REF] is based on first order temporal logic. This proposition is close to our approach, but there are some limits since (i) groups used to compute the support are predefined whereas we consider the fact that the user should be able to define them (see reference dimensions below), (ii) several attributes cannot appear in the sequences. The authors claim that they aim at considering several dimensions but they have only shown one dimension for the sake of simplicity. However, the paper does not provide any complete proposition to extend this point to real multidimensional patterns.

M 2 SP: Mining Multidimensional Sequential Patterns

As shown above, no work has been done in order to mine sequential patterns over several dimensions. The work described in [START_REF] Pinto | Multi-dimensional sequential pattern mining[END_REF] is said to be intra-pattern since sequences are mined within the framework of a single description (the so-called pattern). In this paper, we propose to generalize this work to inter-pattern multidimensional sequences. Given a table T , the set of all tuples in T having the same value r on D R is said to be a block and we denote by B T,D R the set of blocks from table T . Fig. 5 shows blocks built from table T . Each block B in B T,D R is denoted by the tuple r that defines it.

Dimension Partition

In our running example, we consider

F = / 0, D R = {CG,C}, D A = {A, P}, D t = {D}.
When mining multidimensional sequential patterns, the set D R identifies the blocks of the database to be considered when computing the support. For this reason, this set is called reference. The support of a sequence is the proportion of blocks embedding it. Note that usual sequential patterns, and sequential patterns from [START_REF] Pinto | Multi-dimensional sequential pattern mining[END_REF] and [START_REF] De Amo | An apriori-based approach for first-order temporal pattern mining[END_REF], this set is reduced to one dimension (cid in [START_REF] Pinto | Multi-dimensional sequential pattern mining[END_REF] or IdG in [START_REF] De Amo | An apriori-based approach for first-order temporal pattern mining[END_REF]). The set D A describes the analysis dimensions, meaning that these dimensions will be found in the multidimensional sequential patterns. Note that usual sequential patterns only consider one analysis dimension corresponding to the products purchased or the web pages visited. The set F describes the ignored dimensions, which are used neither to define the date, nor the blocks, and which are not present within the patterns mined.

Multidimensional Item, Itemset and Sequential Pattern Definition 1 (Multidimensional Item) A multidimensional item e defined on D

A = {D i 1 , . . . , D i m } is a tuple e = (d i 1 , . . . , d i m ) such that ∀k ∈ [1, m], d i k ∈ Dom(D i k ).

Definition 2 (Multidimensional Itemset) A multidimensional itemset i defined on D

A = {D i 1 ,. . . ,D i m } is a non empty set of items i = {e 1 , . . . , e p } where ∀ j ∈ [1, p], e j is a mul- tidimensional item defined on D A and ∀ j, k ∈ [1, p], e j = e k .
Note that all items from an itemset are defined using the same dimensions (D A ). Also note that all pairs of multidimensional items from an itemset are different.

Definition 3 (Multidimensional Sequence)

A multidimensional sequence ς defined on D A = {D i 1 ,. . . ,D i m } is an ordered non empty list of itemsets ς = i 1 , . . . , i l where ∀ j ∈ [1, l], i j is a multidimensional itemset defined on D A .

Definition 4 (Inclusion of sequence)

A multidimensional sequence ς = a 1 , . . . , a l is said to be a subsequence of a sequence ς = b 1 , . . . , b l if there exist integers

1 ≤ j 1 ≤ j 2 ≤ . . . ≤ j l ≤ l such that a 1 ⊆ b j 1 , a 2 ⊆ b j 2 , . . . , a l ⊆ b j l .
Let ς = {(Y, c, 50)}, {(M, r, 10)} and ς = {(Y, c, 50), (M, p, 2)}, {(M, r, 10)} be two multidimensional sequences. ς is a subsequence of ς .

Support

Computing the support of a sequence amounts to counting the number of blocks that contain the sequence. A block supports a sequence if it is possible to find a set of tuples which satisfy it. For each itemset from the sequence, we must thus find a date from Dom(D t ) such that all items from the itemset appear at this date. All itemsets must be retrieved at different dates from Dom(D t ) such that the order of the itemsets from the sequence is satisfied.

Definition 5 A table T supports a sequence i

1 , . . . , i l if ∀ j = 1 . . . l, ∃d j ∈ Dom(D t ), for every item e in i j , ∃t = ( f , r, e, d j ) ∈ T with d 1 < d 2 < . . . < d l .
The block (Educ, NY ) from Fig. 2 supports {(Y,c 1. block (Educ, NY) (Fig. 2). Two dates can be found in this block. At date 1, we have (Y, c, 50) and (M, p, 2) and at date 2, we have (M, r, 10). Thus this block supports ς. 2. block (Educ, LA) (Fig. 3). This block cannot be counted since it does not contain (M, p, 2). 3. block (Ret., SF) (Fig. 4). This block contains only one date. Therefore, it is not possible ς to be embedded in it. We have thus support(ς) = 1 3 ≥ minsup.

Jokerized Sequential Patterns

Considering the definitions above, an item can only be retrieved if there exists a frequent tuple of values from domains of D A containing it. For instance, it can occur that neither (Y, r) nor (M, r) nor (O, r) is frequent whereas the value r is frequent. Thus, we introduce the joker value * . In this case, we consider ( * , r)which is said to be jokerized.

∀d i = * , δ ∈ Dom(D i ) such that e [d i /δ] is frequent.
A jokerized item contains at least one specified analysis dimension. It contains a * only if no specific value from the domain can be set. A jokerized sequence is a sequence containing at least one jokerized item. A block is said to support a sequence if a set of tuples containing the itemsets satisfying the temporal constraints can be found.

Definition 9 (Support of a Jokerized Sequence) A table T supports a jokerized sequence

ς = i s1 , . . . , i sl if ∀ j ∈ [1, l], ∃δ j ∈ Dom(D t ), ∀e = (d i 1 , . . . , d i m ) ∈ i j , ∃t = ( f , r, (x i 1 , . . . , x i m ), δ j ) ∈ T with d i = x i or d i = * and δ 1 < δ 2 < . . . < δ l . The support of ς is the proportion of blocks containing ς. support(ς) = |{B∈B T,D R s.t. B supports ς}| |B T,D R |

Mining 1-frequent items

1-frequent items are items built on the analysis dimensions. When considering no joker value, a single scan of the database results in the multidimensional items being frequent. When considering jokerized items, a levelwise algorithm is used in order to build the frequent multidimensional items having the smallest possible number of joker values.

To this end, we consider a lattice which lower bound is the ( * , . . . , * ) multidimensional item. This lattice is partially built from ( * , . . . , * ) up to the frequent items containing as few * as possible. At level i, i values are specified. Then items at level i are combined to build a set of candidates at level i + 1. Two frequent items are combined to build a candidate if they are -compatible. Note that this method is close to the one used for iceberg cubes in [START_REF] Beyer | Bottom-up computation of sparse and iceberg cube[END_REF][START_REF] Casali | Cube lattices: A framework for multidimensional data mining[END_REF]. F i 1 stands for the set of 1-frequent items having i dimensions which are specified (different from * ). Frequent items of size 1 are obtained from the candidate items of size 1. More generally, the algorithm to build candidate items of size i are obtained by considering the frequent items of size i -1:

-compatible if ∃∆ = {D i 1 , . . . , D i n-2 } ⊂ {D 1 , . . . , D n } such that for every j ∈ [1, n -2], d i j = d i j = * with d i n-1 = *
. , v n ) where v i = d i if d i = d i , v i = d i if d i = * and v i = d i if d i = * . Let E
F 1 1 = { f ∈ Cand 1 1 , support( f ) ≥ minsup}, Cand i 1 = F i-1 1 F i-1 1 .

Mining Jokerized Multidimensional Sequences

Once 1-frequent items are mined, the candidate sequences of size k (k ≥ 2) are generated and validated to keep the frequent items. This computation is based on usual algorithms such as PSP [START_REF] Masseglia | The PSP Approach for Mining Sequential Patterns[END_REF] by introducing the treatment of joker values. The support of a sequence ς in a table T according to the reference dimensions D R is given by Algorithm 1. The reference dimensions are used to compute the partition to be considered. This algorithm checks whether each block of the partition supports the sequence by calling the function supportTable (Algorithm 2). supportTable attempts to find a tuple from the block that matches the first item of the first itemset of the sequence in order to anchor the sequence. This operation is repeated recursively until all itemsets from the sequence are found (return true) or until there is no way to go on further (return false). Several possible anchors may be tested if the first ones do not fit.

In this section, we report experiments performed on synthetic data. These experiments aim at showing the interest and scalability of our approach, especially in the jokerized approach. As many databases from the real world include quantitative information, we have distinguished a quantitative dimension. In order to highlight the particular role of this quantitative dimension, we consider four ways of computing frequent sequential patterns: (i) no joker (M 2 SP), (ii) jokers on all dimensions but the quantitative one (M 2 SP-al pha), (iii) jokers only on the quantitative dimension (M 2 SP-mu), (iv) jokers on all dimensions (M 2 SP-al pha-mu). Note that case (iv) corresponds to the jokerized approach presented in Section 5. Our experiments can thus be seen as being conducted in the context of a fact table of a multidimensional database, where the quantitative dimension is the measure. In Figures 56789101112, minsup is the minimum support taken into account, nb dim is the number of analysis dimensions being considered, DB size is the number of tuples, avg card is the average number of values in the domains of the analysis dimensions.

Fig. 6 and 7 compare the behavior of the four approaches described above when the support changes. M 2 SP-al pha and M 2 SP-al pha-mu have a similar behavior, the difference being due to the verification of quantities in the case of M 2 SP-al pha. Note that these experiments are not led on the same minimum support values since no frequent items are found for M 2 SP and M 2 SP-mu if the support is too high. Fig. 8 shows the scalability of our approach since runtime grows almost linearly when the database size increases (from 1, 000 tuples up to 26, 000 tuples). Fig. 9 shows how runtime behaves when the average cardinality of the domains analysis dimensions changes. When this average is very low, numerous frequent items are mined among few candidates. On the contrary, when this average is high, numerous candidates have to be considered from which few frequent items are mined. Between these two extrema, the runtime decreases. Fig. 10 and 11 show the behavior of our approach when the number of analysis dimensions changes. The number of frequent items increases as the number of analysis dimensions grows, leading to an increase of the number of frequent sequences. Fig. 12 and 13 show the differential between the number of frequent sequences mined by our approach compared to the number of frequent sequences mined by the approach described in [START_REF] Pinto | Multi-dimensional sequential pattern mining[END_REF], highlighting the interest of our proposition.

Conclusion

In this paper, we propose a novel definition for multidimensional sequential patterns. Contrary to the propositions from [START_REF] De Amo | An apriori-based approach for first-order temporal pattern mining[END_REF][START_REF] Pinto | Multi-dimensional sequential pattern mining[END_REF][START_REF] Yu | Mining sequential patterns from multidimensional sequence data[END_REF], several analysis dimensions can be found in the sequence. This allows the discovery of rules like A customer who bought a surfboard together with a bag in NY later bought a wetsuit in LA. We also define jokerized sequential patterns by introducing the joker value * on analysis dimensions. Moreover, reference dimensions are defined in order to generalize the way clients are defined. The algorithms implementing our approach are given and evaluated with experiments performed on synthetic data. This work can be extended following several directions. For example, we can take into account approximate values on quantitative dimensions. In this case, we allow the consideration of values that are not fully jokerized while remaining frequent. This proposition is important when considering data from the real world where the high number of quantitative values prevent each of them to be frequent. Rules to be built will then be like The customer who bought a DVD player on the web is likely to buy almost 3 DVDs in a supermarket later. Hierarchies can also be considered in order to mine multidimensional sequential patterns at different levels of granularity in the framework of multidimensional databases.

Fig. 1 .

 1 Fig. 1. Table T

Fig. 2 .Fig. 3 . 2 Fig. 4 .Fig. 5 .

 23245 Fig. 2. block (Educ, NY )

  In our running example, (Y, c, 50), (M, p, 2), (M, r, 10) are three multidimensional items on D A = {A, P, Q}. {(Y, c, 50), (M, p, 2)} and {(M, r, 10)} are multidimensional itemsets on D A . {(Y, c, 50), (M, p, 2)}, {(M, r, 10)} is a multidimensional sequence on D A .

Definition 6 (Definition 7 (

 67 , 50), (M, p, 2)}, {(M, r, 10)} since {(Y, c, 50), (M, p, 2)} appears at date = 1 and {(M, r, 10)} appears at date = 2. The support of a sequence in a table T is the proportion of blocks of T that support it. Sequence Support) Let D R be the reference dimensions and T be a table partitioned into the set of blocks B T,D R . The support of a sequence ς is:support(ς) = |{B∈B T,D R s.t. B supports ς}| |B T,D R | Frequent Sequence) Let minsup ∈ [0, 1]be the minimum user-defined support value. A sequence ς is said to be frequent if support(ς) ≥ minsup. Note that an item e is said to be frequent if so is the sequence {e} . Let us consider D R = {CG,C}, D A = {A, P}, minsup = 1 5 , ς = {(Y, c, 50), (M, p, 2)}, {(M, r, 10)} . The three blocks of the partition of T from Fig. 5 must be scanned to compute support(ς).

  and d i n-1 = * and d i n = * and d i n = * . Definition 11 (Join) Let e 1 = (d 1 , . . . , d n ) and e 2 = (d 1 , . . . , d n ) be two -compatible multidimensional items. We define e 1 e 2 = (v 1 , . .

  and E be two sets of multidimensional items of size n, we define E E = {e e s.t. (e, e ) ∈ E × E ∧ e and e' are -compatible} Two multidimensional items defined on n dimensions are -compatible if they share n-2 values. For instance, (NY,Y, * ) and ( * ,Y, r) are -compatible. We have (NY,Y, * ) ( * ,Y, r) = (NY,Y, r). On the contrary, (NY, M, * ) and (NY,Y, * ) are not -compatible.

Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .

 891011 Fig. 6.Runtime over Support (DB size=12000, nb dim=5, avg card=20)

  , and we briefly recall the basic ingredients of this model used in this paper. Let U = {D 1 , . . . D n } be a set of attributes, which we call dimensions in our approach. Each dimension D i is associated with a (possibly infinite) domain of values, denoted by dom(D

i ). A relational table T over universe U is a finite set of tuples t = {d 1 , . . . , d n } such that, for every i = 1, . . . , n, d i ∈ dom(D i ). Moreover, given a table T over U, for every i = 1, . . . , n, we denote by Dom T (D i ) (or simply Dom(D i ) if T is clear from the context) the active domain of D i in T , i.e., the set of all values of dom(D i ) that occur in T .

  Definition 10 ( -compatibility) Let e 1 = (d 1 , . . . , d n ) and e 2 = (d 1 , . . . , d n ) be two distinct multidimensional items where d i and d i ∈ dom(D i ) ∪ { * }. e 1 and e 2 are said to be

Function supportcount Data

: ς, T, D R , counting Result : support of ς Integer support ←-0 ; BooleanseqSupported;


Malik Shukeir

Laurent M Dupuy
email: laurent.dupuy@cea.fr

Benoit Devincre

Simulation of Orowan's mechanism using dislocation dynamics based on the non-singular elastic theory

Keywords: Dislocation dynamics simulation, Orowan mechanism, Non-singular elastic theory

Simulation of Orowan's mechanism using dislocation dynamics based on the non-singular elastic theory

Introduction

Dislocation Dynamics (DD) simulations rely inherently on a continuum description of dislocations based on the linear elasticity (e.g. [START_REF] Kubin | Dislocations, Mesoscale Simulations and Plastic Flow[END_REF]), whose accuracy in capturing the strain and stress fields around dislocations has been proven experimentally over the years (e.g. [START_REF] Hirth | Theory of dislocations[END_REF]). As such, this simulation method nevertheless fails at describing the discrete atomic positions in the immediate vicinity of a dislocation, and consequently its energy. As pointed out by Bulatov and Cai [START_REF] Bulatov | Computer Simulations of Dislocations[END_REF], this remark holds even for non-singular elasticity models [START_REF] Peierls | The size of a dislocation[END_REF][START_REF] Miller | A non-local formulation of the peierls dislocation model[END_REF][START_REF] Cai | A non-singular continuum theory of dislocations[END_REF] where the stress field remains finite in the core of the dislocation. Most DD simulations therefore partition the total energy of a dislocation into an elastic and a core-energy contribution [START_REF] Arsenlis | Enabling strain hardening simulations with dislocation dynamics[END_REF], the latter being added to account for whatever is left by the former in the dislocation core. Despite noticeable efforts in feeding atomistic information to these models [START_REF] Martinez | Atomistically informed dislocation dynamics in fcc crystals[END_REF][START_REF] Geslin | Implementation of the nudged elastic band method in a dislocation dynamics formalism: Application to dislocation nucleation[END_REF][START_REF] Boleininger | Atomistic-to-continuum description of edge dislocation core: Unification of the peierls-nabarro model with linear elasticity[END_REF], limited efforts have been made so far to identify the relevant parameters on atomistic data, nor to assess their influence on classical dislocation mechanisms such as the Orowan process. The objective of this study is therefore to address this issue and provide guidelines to choose the corresponding parameters.

Among the various approaches used in current DD codes ([7, 11, 12, 13, 14, 15]) to account for dislocations total strain energy, the choice was made here to use the isotropic non-singular elastic model of Cai et al. [START_REF] Cai | A non-singular continuum theory of dislocations[END_REF] in combination with a basic core energy model. In the absence of a comprehensive atomistic data base, the latter is simply based on the orientation dependent line tension approach [START_REF] Wit | The Self-Energy of Dislocation Configurations Made up of Straight Segments[END_REF]. The former offers many mathematical and computational advantages, and is currently widely used. As we will discuss it latter, these choices restraint the relevant energy parameters to two values. We therefore believe that this study should be of use and applicable to almost all the DD codes.

The Orowan mechanism [START_REF] Orowan | Discussion on internal stresses[END_REF] was selected in this study as our reference case. This mechanism, which is the bypassing of strong precipitates made by a dislocation line moving in a given glide plane, was preferred to other elementary phenomena for the following reasons. First, it has an important impact on plastic strain hardening [START_REF] Nembach | Particle strengthening of metals and alloys[END_REF][START_REF] Bacon | Dislocation-obstacle interactions at atomic level in irradiated metals[END_REF] and has been widely studied in the literature at the mesoscopic scale using dislocation dynamics [START_REF] Bacon | The effect of dislocation self-interaction on the orowan stress[END_REF][START_REF] Mohles | The peak-and overaged states of particle strengthened materials: computer simulations[END_REF][START_REF] Monnet | Investigation of precipitation hardening by dislocation dynamics simulations[END_REF][START_REF] Queyreau | Orowan strengthening and forest hardening superposition examined by dislocation dynamics simulations[END_REF][START_REF] Monnet | Multiscale modeling of precipitation hardening: Application to the Fe-Cr alloys[END_REF]. Secondly, it does not rely on unphysical arbitrarily pinned dislocation ends by contrast with the Frank-Read source [START_REF] Foreman | The bowing of a dislocation segment[END_REF] and can therefore be simultaneously investigated using molecular dynamics (MD) [START_REF] Bacon | Dislocation-obstacle interactions at atomic level in irradiated metals[END_REF][START_REF] Terentyev | Interaction of a 1 2< 111> screw dislocation with Cr precipitates in bcc Fe studied by molecular dynamics[END_REF][START_REF] Granberg | Interaction of dislocations with carbides in BCC Fe studied by molecular dynamics[END_REF]. Finally, the Orowan mechanism questions simultaneously the underlying dislocation strain energy through its effective line tension and the stress required to bow out the side arms of the dislocation, as well as its stress field and the corresponding elastic interaction between the arms as the dislocation bypasses the precipitates [START_REF] Bacon | The effect of dislocation self-interaction on the orowan stress[END_REF]. Previous studies investigated the effect of precipitate size and distribution on the mechanical properties, but to our knowledge no study investigated in a systematic manner the influence of the dislocation strain energy parameters used in DD simulations.

This paper is organized as follows. In the next section we present a brief overview of dislocation elastic theory and parameters appearing in the dislocation strain energy definition used in most DD simulations, as well as the standard model used to predict Orowan stress in the case of impenetrable obstacles. In section (3), we describe the details of the parametric study performed in this article. The corresponding results are presented in section (4). The last section is dedicated to a discussion and concluding remarks.

Theoretical background

In linear theory of elasticity, it is convenient to split the total energy of a dislocation into two separate terms, one for the long range elastic field and the other for the core energy [START_REF] Hirth | Theory of dislocations[END_REF].

E total = E elastic + E core (1)
Most of the total energy of a dislocation comes from the elastic strain energy contribution, while the core energy is reported to constitute a few percent of the total elastic strain energy [START_REF] Hull | Introduction to dislocations[END_REF][START_REF] Kubin | Dislocations, Mesoscale Simulations and Plastic Flow[END_REF]. Assuming isotropic elasticity, the self-energy per unit length of a dislocation stored in a cylindrical ring of inner radius r 0 (core radius) and outer radius R 0 in the case of infinite straight dislocations is

E in f elastic = µb 2 4π(1 -ν) (1 -ν cos 2 θ) ln( R 0 r 0 ) ( 2
)
where µ is the isotropic shear modulus, b is the Burgers vector and ν is the Poisson's ratio. The dislocation character term θ is the angle between the Burgers vector and the tangent vector along the dislocation line. From Equation (2), we simply see that the elastic energy of a dislocation depends on i) the dislocation core definition via r 0 , ii) the initial and boundary condition via the definition of R 0 and iii) the dislocation character (edge or screw) via the pre-logarithmic energy constant [START_REF] Hirth | Theory of dislocations[END_REF][START_REF] Kubin | Dislocations, Mesoscale Simulations and Plastic Flow[END_REF].

Because of the inability of the elastic theory to represent the core energy of a dislocation, and in the absence of comprehensive atomistic information, the choice was made here to define the core energy as being directly proportional to the elastic energy per unit length of an infinite straight dislocation through a single parameter, α core , such as:

E core = α core E in f elastic (3)
This approach is consistent with several available atomistic studies (e.g. [START_REF] Monnet | Structure and mobility of the 12¡ 111¿{112} edge dislocation in bcc iron studied by molecular dynamics[END_REF][START_REF] Kubin | Dislocations, Mesoscale Simulations and Plastic Flow[END_REF]) in which the core energy amounted to a few percent of the elastic energy. For convenience, we introduce at this point a term called the core energy parameter ζ core . The latter constant quantity take the form:

ζ core = α core ln( R 0 r 0 ) (4)
The core energy can therefore be written as:

E core = ζ core µb 2 4π(1 -ν) (1 -ν cos 2 θ) (5)
The driving force controlling dislocation dynamics is mainly a function of the loading stress, and internal stresses generated by each dislocation line. The calculation of the internal stress field associated with an ensemble of curved dislocations needs integrating equation [START_REF] Hirth | Theory of dislocations[END_REF][START_REF] Devincre | Three dimensional stress field expressions for straight dislocation segments[END_REF][START_REF] Cai | A non-singular continuum theory of dislocations[END_REF]:

σ αβ = µ 8π ∂ i ∂ p ∂ p R[b m ε imα dx β + b m ε imβ dx α ] + µ 4π(1 -ν) b m ε imk (∂ i ∂ α ∂ β R -δ αβ ∂ i ∂ p ∂ p R)dx k (6)
where R is the distance between points α and β on the dislocation line and i jk is Levi-Civita notation.

A major difficulty in calculating the stress field of dislocations is related to the fact that Equation ( 6) is divergent when R approaches zero. Different solutions have been proposed to eliminate such singularity as in [START_REF] Miller | A non-local formulation of the peierls dislocation model[END_REF][START_REF] Cai | A non-singular continuum theory of dislocations[END_REF][START_REF] Po | A non-singular theory of dislocations in anisotropic crystals[END_REF]. Among those solutions, the one proposed by Cai et al. [START_REF] Cai | A non-singular continuum theory of dislocations[END_REF] features some specifications which makes it convenient for DD simulations. In brief, it modifies the singular stress field solution through a mathematical transformation where R in Equation ( 6) is replaced by R a = R 2 + a 2 0 . This transformation implies the definition of a new regularization parameter a 0 coined as the dislocation core width parameter.

Hence, the dislocation strain energy definition used in this study reduces to two major parameters: the dislocation core width parameter a 0 and the core energy parameter ζ core . In order to assess the impact of these parameters, we perform DD simulations on the Orowan mechanism.

In a seminal work, Bacon et al. [START_REF] Bacon | The effect of dislocation self-interaction on the orowan stress[END_REF] investigated the bypass of a periodic row of impenetrable spherical obstacles by an infinite dislocation with the Orowan's mechanism [START_REF] Orowan | Discussion on internal stresses[END_REF]. From an analysis of the influence of the obstacle size (D) and the inter-obstacle distance (L) on the flow stress, they defined the following equation (hereinafter referred to the BKS model):

τ = A µb L [ln D + B] (7)
In Equation ( 7), A is a pre-logarithmic factor that depends on the character of the dislocation. A = 1 and A = 1/(1 -ν) for edge and screw dislocations, respectively. B is a fitting parameter evaluated to 0.7 in the BKS paper. This model equation is widely used in the literature as a reference for Orowan-like interactions [START_REF] Queyreau | Orowan strengthening and forest hardening superposition examined by dislocation dynamics simulations[END_REF][START_REF] Monnet | Multiscale modeling of precipitation hardening: Application to the Fe-Cr alloys[END_REF].

Simulation technique

Simulations were carried out using NUMODIS [15,[START_REF] Li | Screw dislocation interaction with irradiation defect-loops in α-iron: evaluation of cross-slip effect using dislocation dynamics simulations[END_REF], a 3D nodal dislocation dynamics (DD) code based on the isotropic elastic theory of dislocations. Dislocation lines are represented by a set of nodes, interconnected by straight segments nodes, whose properties are their Burgers vector and their glide plane.

This study is performed on a single crystal of BCC Iron at 300K. At this temperature, the lattice parameter equals 0.2855 nm, b the Burgers vector of slip systems 1 / 2 111 (11 0) equal 0.2475 nm and the shear modulus µ equal 63 GPa. The x, y and z axes of the simulated volume are oriented in the [111], [ 11 2] and [11 0] crystallographic directions, respectively. Periodic boundary conditions are applied in the [111] and [ 11 2] directions, while no particular conditions are applied to the boundary surfaces in the [11 0] direction since they are normal to the dislocation glide direction.

Following a robust methodology previously used to simulate the Orowan process [START_REF] Monnet | Multiscale modeling of precipitation hardening: Application to the Fe-Cr alloys[END_REF][START_REF] Queyreau | Orowan strengthening and forest hardening superposition examined by dislocation dynamics simulations[END_REF][START_REF] Keyhani | An efficient computational technique for modeling dislocation-precipitate interactions within dislocation dynamics[END_REF], one 1 / 2 111 (11 0) edge or screw dislocation is introduced in the periodic volume in front of an impenetrable spherical obstacle with diameter D and cut by the dislocation glide plane at its center (Figure 1). This obstacle is considered as an incoherent inclusion, whose elastic properties are equal to the surrounding crystal. No specific stress field is therefore associated with the obstacle. The spacing L between periodic images of the obstacle is varied by changing the size of the simulation box along the direction parallel to the dislocation line. The dimension of the simulated volume in the glide direction is systematically adjusted to allow for Orowan loop formation before the dislocation line reach the boundary of the periodic volume. The z-axis is systematically adjusted to be three times the obstacle diameter. Pure shear stress is applied in the b direction to impose a constant strain rate which corresponds to a constant dislocation velocity in the glide plane close to 3m/s. For reasons of simplicity and to allow for comparison with previous computations made in FCC materials, we consider in the simulations a simple linear over-damped mobility law similar to [START_REF] Shi | Interaction of ¡100¿ dislocation loops with dislocations studied by dislocation dynamics in α-iron[END_REF], in the form:

v s = τb B ( 8
)
where τ is the effective resolved shear stress and B is a viscosity coefficient set to 8 * 10 -5 Pa.s. The latter quantity accounts for dissipating processes like dislocation-phonon interactions.

All simulations are done in a quasi-static condition and tests have been made to verify that the viscosity coefficient value has no influence on the computed critical stress. Here, it must be noted that although thermally activated bypassing mechanisms are reported in the literature in case of small obstacles, for reasons of simplicity no thermally activated dislocation property like, dislocation cross-slip or climb, are considered in the present simulations. All the material parameters used in the DD simulations are consistent with prior molecular dynamics simulations found in [START_REF] Terentyev | The effect of temperature and strain rate on the interaction between an edge dislocation and an interstitial dislocation loop in α-iron[END_REF][START_REF] Terentyev | Simulation of the interaction between an edge dislocation and a < 100> interstitial dislocation loop in α-iron[END_REF][START_REF] Queyreau | Edge dislocation mobilities in bcc Fe obtained by molecular dynamics[END_REF].

In order to investigate the influence of the two core parameters, a parametric study of the Orowan mechanism was conducted. The range of the many combinations we tested are summarized in Table [START_REF] Kubin | Dislocations, Mesoscale Simulations and Plastic Flow[END_REF].

Simulation results

The results of all the calculations of the Orowan critical stress we conducted are plotted in Figure [START_REF] Hirth | Theory of dislocations[END_REF]. For comparison with the BKS model, the Orowan stress is plotted as a function of the harmonic mean of the inter-obstacle spacing and obstacle diameter, (L -1 + D -1 ) -1 . A large dispersion of the values is found when changing the parameters controlling the dislocation elastic energy. In the edge dislocation case, and for different values of the harmonic mean, an increase of 100% in the critical stress is observed. Such dispersion is found to be even more important Figure 2: Result of the Orowan stress calculation with screw and edge dislocations in reduced units. Each mark is a unique combination of the parameters reported in Table [START_REF] Kubin | Dislocations, Mesoscale Simulations and Plastic Flow[END_REF]. Continuous lines are the prediction of the BKS model (Equation 7).

To understand the effect of each parameter, our simulation results are now plotted separately by fixing all but one of the energy parameters. In The critical stress of an edge dislocation is observed to be quasi-independent of the Poisson ratio ν, while a non-linear dependence of the form 1/(1-ν) is observed in the case of a screw dislocation. This observation is in agreement with the line tension model proposed by De Wit [START_REF] Wit | The Self-Energy of Dislocation Configurations Made up of Straight Segments[END_REF] that accounts for the effect of the dislocation character and the simulation results first reported in [START_REF] Bacon | The effect of dislocation self-interaction on the orowan stress[END_REF]. Figure [START_REF] Bulatov | Computer Simulations of Dislocations[END_REF] as well shows that when ν equals zero, the Orowan stress is independent of the dislocation character and the Orowan stress is independent of the dislocation line character as expected.

The second analyzed simulation parameter is the core energy parameter ζ core . As shown in Equation (3), α core has a linear influence on the total strain energy of the dislocation and therefore on dislocation line tension. Hence, if the ratio ln(R 0 /r 0 ) is constant, a linear dependence is observed on the Orowan stress as function of ζ core . The same tendency is observed on screw and edge dislocations (see Figure 4). More precisely, increasing ζ core by a factor of 10 increases the critical stress by about 33%. This reveals a fairly high dependence of DD simulations on the definition of the dislocation core energy. This point has been probably been overlooked in several existing studies. The inversely logarithmic dependence observed in Figure ( 5) can be motivated by the dislocation energy expression computed by Cai et al. [START_REF] Cai | A non-singular continuum theory of dislocations[END_REF] in the context of the non-singular theory. Indeed, the core width parameter a 0 appears in the denominator of the logarithmic term, as if r 0 was replaced by a 0 in equation [START_REF] Hirth | Theory of dislocations[END_REF]. Consequently, the simulated Orowan stress is decreased by approximately 18% and 16%, for screw and edge dislocations respectively, when increasing the regularization parameter a 0 by a factor of 3. This result raises doubts about the relevance of DD simulations, which use regulation parameters 10 to 100 times larger than the core radius of the dislocations defined by atomic simulations for computational reasons.

Lastly, we show that in agreement with the BKS model for a given set of simulation parameters, the Orowan stress is logarithmically dependent on the harmonic mean of inter-obstacle distance and obstacle size. Such result is presented in Figure ( 6) with a set of simulation parameters (a 0 = 4.5 nm, ζ core = 0.86 and ν = 0.33) and is in good agreement with the solution found in previous simulations. As discussed in [START_REF] Bacon | The effect of dislocation self-interaction on the orowan stress[END_REF], the evolution of the Orowan stress is here well described with the help of a harmonic mean between the inter-obstacle distance L and the obstacle diameter D. When L is much larger than D the average tends to L and the required stress to overcome the precipitate decreases, since the line tension of a dislocation in inversely proportional to its length, and vice versa.

Discussion and concluding remarks

In this work, we studied in α-iron the interaction of infinite 1 / 2 111 (11 0) screw and edge dislocations with a periodic array of impenetrable obstacles of different size and spacing. These configurations are standard simulation problems previously used to study the Orowan mechanism with MD and DD simulations. The reported results are in good agreement with previous studies, but reveal a large dispersion of results when changing the simulation parameters used to define the dislocation strain energy. More specifically, for the different solutions of simulation parameters, the calculated Orowan stress can vary by 100% and 280% for edge and screw dislocations, respectively.

In order to reveal the collective effect of the studied parameters to the Orowan stress, we propose to combine them in a new equation. This equation represents the sum of two contributions. It reflects the strain energy decomposition into elastic and core energy contributions (see Equation 1). The first contribution is similar to the one defined in the BKS model, but includes the contribution of the core-width parameter a 0 , as it modifies the elastic strain energy in the framework of the non-singular dislocation theory. The second term accounts more specifically for the contribution of the dislocation core energy via the parameter ζ core . Indeed, as revealed in Figure [START_REF] Peierls | The size of a dislocation[END_REF], this contribution contributes significantly to the dislocation line tension and cannot be neglected when modeling phenomena involving dislocations curvature like the Orowan process. The following modified Orowan equation is the outcome of fits made with more than 1000 simulations run with different parameters.

τ Orowan = µb L A 2π [1.23( ln D a 0 -0.18 )+ζ core (9)
The harmonic mean term of L and D appears in the numerator in conformity with the BKS model, while the core width parameter (a 0 ) appears in the denominator since τ Orowan has an inverse dependence on this term. The additional right hand side term of the equation account for the linear contribution of the core energy to the Orowan process. As illustrated in Figure [START_REF] Arsenlis | Enabling strain hardening simulations with dislocation dynamics[END_REF], prediction made with Equation ( 9) gives excellent results and a correlation factor better than 0.99 when considering all our simulation data. 9) prediction for the 1000 combinations of parameters taken from Table [START_REF] Kubin | Dislocations, Mesoscale Simulations and Plastic Flow[END_REF] and tested in the present study.

It should be noted that Equation ( 9) is generic and can be used to interpolate any type of results on the Orowan mechanism. Thus, it becomes possible to perform a reverse analysis to identify the parameters of a DD simulation that can reproduce experimental data or other simulation results. In the following, an example of such adjustment is given. Equation ( 9) is used to define the parameters that are needed in our DD simulation to reproduce the results on the Orowan stress obtained by Lehtinen et al. with MD simulations [START_REF] Lehtinen | Multiscale modeling of dislocation-precipitate interactions in Fe: From molecular dynamics to discrete dislocations[END_REF]. These results are discussed and compared to the BKS model. The results of DD simulations when parametrized with the help of Equation ( 9) show an excellent match with the Orowan stresses computed with MD simulations. More precisely, the difference between the stress found in DD and MD simulations at different obstacles spacing is on average less than 2.4%. It is worth noting also that this difference is 53% between the BKS model and the MD simulation results. Such a difference could be interpreted as coming from the existence of thermally activated phenomena in the MD simulations that cannot be reproduced with any approach based only on the elastic theory. Our study, rather suggests that the dislocation strain energy that controls the dislocation dynamics in the MD simulations is simply different from the definition used initially in the BKS model.

To conclude, it should be noted that the parameterization of the dislocation strain energy in the framework of the non-singular dislocation theory is governed by the idea that the total dislocation energy should remain unchanged when changing the regularization amplitude. Hence, when increasing the dislocation core width parameter in a DD simulation (to avoid using tiny time steps), the dislocation core parameter is usually also increased to keep the total energy constant. As illustrated by our study, such solution without a systematic investigation of the parameterization effect could be problematic. Indeed, we show that the dislocation dynamics are not affected in a simple manner by both parameters. For instance, some new DD simulations to be published in a forthcoming paper have shown that when modeling the interaction between a dislocation and radiation-induced loop defects, a drastic effect is observed on the dislocation dynamics by changing a 0 a few percent. Such parameterization change may even lead to the modeling of very different contact reactions.

In summary, a large dispersion is observed in DD simulation results depending on the choice of parameters used to control the dislocation strain energy. Such uncertainty has to be eliminated so that DD simulation results can be compared with MD and DD simulations or even experiment. A parametric study is proposed to incorporate the effect of essential simulation parameters in an equation form useful to predict the Orowan stress. This model reveals the existing coherence between the many simulations we performed and provides the means to calibrate rigorously future DD simulations. The model was tested and validated by a direct comparison with MD simulations. This work opens the door to more quantitative comparison between DD simulations and other simulation technics.

Figure 1 :

 1 Figure 1: Illustration of the DD Simulation volume. The green surface represents the glide plane of the mobile dislocation, while arrows at the top and bottom surfaces indicate the direction of the applied shear stress. The straight dislocation represents the initial dislocation configuration while the curved one is the critical configuration observed before the formation of the Orowan loop.

Figure ( 3 )

 3 , we show the effect of changing the Poisson ratio while the other simulation parameters are L = 1000b (247 nm), D = 100b (24.7 nm), a 0 = 0.4 nm and ζ core = 0.86.

Figure 3 :

 3 Figure 3: Variation of the Orowan stress as function of the Poisson ratio for screw and edge dislocations. Other simulation parameters are L = 1000b (247 nm), D = 100b (24.7 nm), a 0 = 0.4 nm and ζ core = 0.86.

Figure 4 :

 4 Figure 4: Variation of the Orowan stress as function of the core energy parameter ζ core . Other simulation parameters are set to L = 100b (24.7 nm), D = 10b (2.47 nm), a 0 = 0.15 nm and ν = 0.435.

Figure 5 :

 5 Figure 5: Variation of the normalized Orowan stress as function of core width parameter a 0 . Other simulation parameters are set to L = 1000b (2470 Å), D = 100b (247Å), ζ core = 0.86 and ν = 0.435.

Figure 6 :

 6 Figure 6: Normalized Orowan stress vs the harmonic mean of inter-obstacle length and obstacle diameter. The simulation parameters controlling the elastic energy are set to a 0 = 4.5 nm, ζ core = 0.86 and ν = 0.33. The continuous lines are the BKS model prediction with the fit parameter calculated in [20].

Figure 7 :

 7 Figure 7: Comparison between simulation results and Equation (9) prediction for the 1000 combinations of parameters taken from Table[START_REF] Kubin | Dislocations, Mesoscale Simulations and Plastic Flow[END_REF] and tested in the present study.

  Lehtinen et al. studied the interaction between 1 / 2 111 (11 0) edge dislocation with noncoherent cementite precipitate (Fe 3 C) of different spacing using molecular dynamics. These calculations were produced using the interatomic potential H13 proposed by Henriksson et al. describing the FeCrC system[START_REF] Henriksson | Atomistic simulations of stainless steels: a many-body potential for the fe-cr-c system[END_REF]. Parameters that can be directly shared between MD and DD simulations areµ = 75 GPa, ν = 0.379 at T = 750K, b = 0.2502nm and obstacle diameter D = 2nm. Use is then made of Equation (9) to identify the missing parameters needed in the DD simulations to correctly fix the dislocation strain energy and to be in agreement with the prediction of the interatomic potential used in the MD simulations. Such adjustment gives for the dislocation core width parameter a 0 = 4.5nm and for the dislocation core energy parameter ζ core = 0.51. Both parameters value are physically meaningful and allow running DD simulations to either reproduce quantitatively the MD simulation results in a few seconds or to upscale the MD simulation results. To illustrate this point, a comparison is made in Figure (8) between the initial results of Lehtinen et al., the results we obtained with the adjusted DD simulation and the BKS model predictions.

Figure 8 :

 8 Figure 8: Comparison between critical Orowan stress as function of the inter-obstacle distance L for obstacle of D = 2 nm found using MD and BKS model and DD model in Equation (9). DD simulation parameters are a 0 = 4.5 nm, ζ core = 0.51.

Table 1 :

 1 Energy and geometrical simulation parameters we explored to calculate the Orowan stress.

	Parameter	Symbol	Range	Unit
	Regularization parameter	a 0	1.5, 2.5, 4.0, 5.0	Å
	Core energy parameter	ζ core	0.09, 0.43, 0.86, 1.28	-
	Poisson ratio	ν	0.0, 0.2, 0.33, 0.435, 0.495	-
	Inter-obstacle distance	L	100.0, 316.2, 1000.0	b
	Obstacle size	D	10.0, 31.62, 100, 316.2	b

in the case of screw dislocations, where the critical stress increases of approximately 280% for different sets of parameters.

Acknowledgments

This work was performed in the framework of SOTERIA project funded by the European Union's "Horizon 2020" research and innovation program under agreement GA number 661913. The authors wish to thank Dr. Ghiath Monnet (EDF, Les Renardièrs) for the fruitful discussions.


Michael V Cotta- Schönberg

Silvio Enea

; Piccolomini

Sylvius Aeneas

Silvius Aeneas

Piccolomini

Michael V Cotta-Schönberg

In December 1459, when the business of the Congress of Mantua was almost concluded, Margrave Albrecht Achilles of Brandenburg finally arrived. He was Pope Pius II's favourite German prince.

Foreword

In 2007, I undertook a project of publishing the Latin texts with English translations of the orations of Enea Silvio Piccolomini / Pope Pius II. Altogether 80 1 orations (including papal responses to ambassadorial addresses) are extant today, though more may still be held, unrecognized, in libraries and archives.

At a later stage the project was expanded to include ambassadors' orations to the pope, of which about 40 are presently known.

I do not, actually, plan to publish further versions of the present volume, but I do reserve the option in case I -during my future studies -come across other manuscripts containing interesting versions of the oration or if important new research data on the subject matter are published, making it appropriate to modify or expand the present text. It will therefore always be useful to check if a later version than the one the reader may have previously found via the Internet is available.

I shall much appreciate to be notified by readers who discover errors and problems in the text and translation or unrecognized quotations.

12 September 2019 MCS 1 81 orations, if the "Cum animadverto" is counted is a Piccolomini-oration, see oration "Quam laetus" [18], Appendix In late December 1459, when the business of the Congress of Mantua was almost concluded, Margrave Albrecht Achilles of Brandenburg finally arrived in Mantua. He was no stranger to the princely court of Mantua since his niece, Barbara, was the spouse of Margrave Ludovico Gonzaga. Albrecht was Pope Pius II's favourite German prince and one whom he thought might be an excellent commander of a joint European army against the Turks. 2

In his Commentarii, Pius wrote briefly about him and his visit, adding:

When he came to Mantua, the Cardinal of San Pietro 3 left the ranks 4 and went out to meet him.The pope received him at a public audience and showered him with praise for his prompt and generous promises of support against the Turks. He also presented him with a thousand ducats and two horses from Puglia. As part of the festivities for Epiphany, he awarded him the sword which was traditionally granted to the most worthy secular prince present at Christmas, as well as a cap adorned with pearls. There is no doubt, that Albrecht's appearance at the Congress of Mantua was a great comfort to the pope, and it possible that Albrecht would have accepted command of a German crusade army had one been formed, but it was not, neither in Mantua nor later.

1 CO,III,45 (Meserve,II,; Picotti, pp. 292-293; Voigt, IV, pp. 104-105 2 Pastor, II, p. 68 3 The German Cardinal Nikolaus von Kues 4 "extra ordinem": instead of "left the ranks", I should prefer the translation "as an extraordinary gesture" / MCS 5 CO, III,45 (Meserve,II, The pope received Albrecht in a public meeting (publico auditorio), where the Dean of Bamberg, Hertnidt vom Stein, gave the oration "Non ullam"1 on Albrecht's behalf, and the pope replied with the oration "Advenisse te citius". Later, on the Feast of Epiphany, the pope gave the margrave the papal sword, traditionally bestowed at Christmas on a secular prince. This may indicate that Albrecht arrived after Christmas day, otherwise the sword would traditionally have been granted to him on that day. That the public reception and the bestowal of the papal sword were separate events, is shown by the wording in the Commentarii: Pontifex in auditorio publico eum excepit laudavitque magnificis verbis2 … In festo vero Epiphaniae … ensem … et pilleum … inter solemnia dedit.

Themes

In the oration the pope expressed his great satisfaction at the margrave's arrival.

He claimed that the margrave's energy and leadership skills were such that he could personally ensure that there would be peace in Germany, allowing the promised German army to be gathered.

He complimented him on his military skills, acquired and demonstrated in many wars.

But such wars, against other Christians -the pope said -necessarily brought guilt for the crimes of war, slaughtering of innocents, rape of women, burning of monasteries etc.

All these sins would be expiated if the Margrave joined the crusade.

Date, place, audience and format

The oration was probably delivered in the last week of 1459.

According to the Ottobon. Lat. 905, f. 52r, the oration on behalf of Albrecht was delivered by Hertnidt vom Stein after the solemn mass on the Feast of Epiphany, 6 January 1460, i.e. in connection with the bestowal of the papal sword. But since the pope's own description of the events in the Commentarii (see above) clearly distinguishes between the reception of the margrave in a consistory where he praised him, evidently in response to an ambassadorial address on behalf of the margrave, either this dating is erroneous, or the pope's memory may have faltered. For the purpose of the present edition, the date of late December is retained, and the date of Hertnidt's oration in the Ottobon. 951 is considered to be erroneous.

The audience consisted of the cardinals, ambassadors, and members of the papal court present in Mantua and other participants in the congress.

The format was a papal oration from the throne, in reply to an address made on behalf of the margrave (Probamus, quae tuo nomine dicta sunt, sect. 1)

Text 1

The oration is included in a special Collection of Responses of Pope Pius II (1460) 2 largely consisting of responses to addresses by ambassadors, presently known to be extant in five manuscripts:

4.1. Manuscripts 3 The Chisianus is the eldest of the five and identical with or very close to the oration as actually delivered by Pius.

Editions

The oration was published by Mansi twice:

• Mansi, Giovanni:

Present edition

For principles of edition (incl. orthography) and translation, see Collected Orations of Enea Silvio Piccolomini / Pope Pius II, vol. 1, ch. 9-10.

Text:

The present edition is based on the manuscripts from Lucca, the Archivio Segreto Vaticano and the Chisianus, with the Chisianus as the lead manuscript.

Pagination:

Pagination is from the lead manuscript.

Sources 1

In this oration, three direct or indirect quotations have been identified: one from the Bible, one from a classical source (Homer), and one from a medieval source (Innocent III).

Bibliography

Annales [2] The glorious and mighty German nation, from which you come, has promised 42.000 soldiers to the expedition for the Christian religion against the Turks. We consider two things to be necessary for the fulfilment of this promise: the first is the stability of the Peace of Nürnberg from last year, and that a new conflict be not allowed to break out. Otherwise an army cannot be mobilized. The second is that great energy and careful diligence be shown in gathering the troops. Both depend on you. If you will, all will be at peace. And if you show diligence, a strong army against the Turks will not be lacking. We know your energy and your diligence: if you apply your mind, all -and even the most difficult -things will happen [as you wish]. Therefore, act, apply yourself to this task, and avail yourself of those things that can make you happy and immortal. [3] We believe that the wars you have conducted were just, but still you have been fighting Christians which easily leads to sin. Often the blood of innocents is shed, and girls and married women are raped. While the wicked glory, the poor are burnt, churches are robbed, monasteries set on fire, and the divine is getting mixed up with the human. For victors the thirst for revenge is more important than moderation. So, We do not doubt that many things must be burdening your conscience: you could not avoid harming many.

[4] But all these [sins] you may easily expiate through the Turkish war. Until now you have been fighting for the world: now fight for Christ. You have been fighting for the body: now fight for the soul. You have been fighting for a fragile and vain glory: now fight for immortal and eternal honour.1 Exert yourself in this war in which you may wash away all the stains of old guilt and make yourself pleasing to your Lord. Do act, and do generously accept this crusade. If We see you armed for it, We shall have great hopes for victory. For We hope that you will be another David in battle, about whom they sang that Saul slew his thousands, and David his ten thousands. 2 Your person will count as much as a great army. We shall consider you to be another Judas Machabee or another Samson. So, be brave and strong, and when you return to your brothers, the German nobles, please endeavour to settle internal conflicts, if any are still festering, and to gather the promised army in Germany and lead it against the enemies. In his mercy, the Lord God will make all else go well. Amen. Among the other orations of Hertnidt vom Stein are his De rectoralis dignitatis acceptatione et pro ipsesi honoris gratiarum actione (Thumser,p. 19), and an oration to Emperor Friedrich III in the matter of the succession conflict in Stettin (Thumser,p. 193). [8] Praeclara haec sunt et infinita, quae de ingenti virtutum tuarum ac meritorum acervo enervari possent, quae subticenda mihi potius esse censeo, quam ex multis pauca in modum referre, ne tenuiter referendo odiosus me potius reddam quam attentum. Hoc jure me dixisse velim pro singularissimis virtutibus tuis et mira in omnes pietate.

Appendix: Oration

Oratio habita per venerabilem et eximium doctorem dominum

[9] Se 2 satis atque satis obnoxium tibi profiteri, testis enim mihi potest esse tota Germania, quanto labore, industria ac diligentia insudaveris 3 . Ac compositis quamplurimis Germaniae partium dissidiis ipsa quieta esset et tibi, ut te exoptare cognoscebat catholico more oboedientiam reddere non denegaret.

[10] Testes mihi praeterea esse possunt oratores tui, quos cum bellorum furor insonaret et quadrupedante putrem sonitu quateret ungula campum 4 nostris in regionibus esse voluisti, qui luce clarius cernere poterant non absque gravi jactura sua cum paci consultasse, cum si bellum illud prosecutus fuisset 5 , se victore palmam in manibus habere cognoscent. Videbat enim bellum illud, quod tum imminebat, tuae deliberationi impedimento fore. Quare tamquam oboediens filius paci manum imposuit furentemque bello Alamaniam totam quiescere cooperatus est.

[11] Testis est mihi -quod quidem potissimum arbitror -praesentia sua. Non enim tot aut frigora aut venti aut pluviae aut nives aut tot bellorum discrimina eum tenuere quin, ut toto affectabat animo, te peteret.

[12] Veram itaque, alme pontifex, morem, qui magis commendari tibi quam laudari voluit. Accipe igitur principem ipsum ex innata bonitate et commendandum in sedem tuam et devotionem et operibus fidelem et multa virtute approbatum, qui postquam a tua sanctitate accersitus ad hanc urbem extitit, etsi parum utilitatis praesentia sua afferre cognosceret ad eam rem, quam se consequi arbitrabatur. Voluit tamen 6 inter ea impedimenta et varia pericula quibus hactenus 1 one word illegible 2 Albrecht von Brandenburg 3 insudaverit cod. 4 Vergilius: Aeneis, 8.596 5 Albrecht von Brandenburg 6 tantum cod.

Table of volumes in Collected Orations of Pope Pius II. 12 vols. Edited and translated by Michael von Cotta-Schönberg Table of contents

	I.	INTRODUCTION
	1.	Context
	2.	Themes
	3.	Date, place, audience, and format
	4.	Text
		4.1. Manuscripts
		4.2. Editions
		4.3. Present edition
	5.	Sources
	6.	Bibliography
	7.	Sigla and abbreviations
	II.	TEXT AND TRANSLATION

Appendix: Oration "Non ullam" of Hertnidt vom Stein to Pope Pius II (6 January 1460, Mantua) I. INTRODUCTION 1. Context 1

• Lucca / Biblioteca Capitolare Feliniana 544

	Barb. lat. 1499, ff. 84r-86r
	Barb. lat. 1602, ff. 106r-108v
	Chis. J.VII.251, ff. 181v-182r (H) *
	, ff. 182v-183v (G) *
	• Archivio Segreto Vaticano
	Arm. XXXII 1, ff. 69r-71r (J) *
	• Roma / Biblioteca Apostolica Vaticana
	1 Concerning the textual transmission of Pius II´s orations, see Collected orations of Pope Pius, vol. 1, ch. 5
	2 See Collected Orations of Pope Pius II, vol. 1, sect. 5.1.2.
	3 Manuscripts for which an orthographical profile is given in Collected orations of Pope Pius II, vol. 11, are marked
	with an asterisk

  Orationes politicae et ecclesiasticae. Ed. Giovanni Domenico Mansi. 3 vols. Lucca: 1] {181v} Advenisse te citius, dilectissime fili, et optavissemus quidem, et rebus gerendis nostro judicio non parum conduxisset. Sed curasti 1 , ne bellum civile inter Alemannos exoriretur: non minus reipublicae profuisti, quam si hic affuisses. Venisti tamen modo et omnes difficultates superasti. Nulla te potuerunt impedire impedimenta, quominus nostro desiderio 2 satisfaceres. Neque fluminibus, neque montibus retineri vera caritas 3 potest. Et praesentia tua et verbis tuis mirum in modum nos recreas. Probamus, quae tuo nomine dicta sunt, et consilia tua non aliter quam viri prudentis et qui multa viderit et probaverit, amplectenda esse ducimus, de quibus tecum seorsum agemus.[2] Verum quia natio germanica, ex qua tu ortus es, inclyta et potens, in 4 expeditionem contra Turcos obeundam duo et XL 5 millia pugnatorum Christianae religioni pollicita est, ut hoc fiat, quod promissum est, duo necessaria esse videmus: unum, ut pax proximo anno Norimbergae 6 composita stabilis sit, {182r} et nihil novi discidii 7 suboriri sinatur, neque enim aliter educi exercitus posset. Alterum, ut ad cogendas copias ingens studium et diligentia solers adhibeatur. Utrumque in te situm est. Si voles, pacata erunt omnia, nec deerit validus contra Turcos exercitus, ubi tua solertia incubuerit. Novimus studium et diligentiam tuam: omnia fiunt quamvis difficillima, ubi tu 8 animum intenderis. Age igitur, erige mentem, et his te rebus accinge, quae te possunt beatum et immortalem reddere. Multa hactenus bella gessisti: in Franconia, quae tibi patria est, in Misna, in Saxonia, in Bohemia, in Slesia, in Polonia, in Hungaria, in Austria, in Bavaria, in Svevia: nulla regio in Germania est tui non plena sudoris. A teneris, ut ajunt, unguiculis, versatus in armis adeo clarum gessisti militem et deinde ducem, ut non ab re Theutonicum te Achillem quidam nominaverunt. Tulisti saepe optima ex hoste spolia. Saepe vallum, saepe murum primus hostilem ascendisti. Erexisti pluribus in locis victoriarum trophaea, et ex manubiis 9 hostium tua templa ditasti. Si cujuspiam nostri temporis viri victura est fama, tua profecto victura est. Injuriam enim scriptores tuo nomini fecerint, nisi te priscis aequaverint, quos Graecia et quos Roma miratur 10 , belli ducibus. Beloved son, We had really hoped that you would arrive earlier, and We believe that it would have greatly facilitated the matter at hand. But since you were endeavoring to prevent a civil war in Germany, you have benefited the [Christian] Commonwealth as much as if you had been present here. But now you have arrived and surmounted all difficulties. No barriers could prevent you from fulfilling Our wish. True charity cannot be held back by rivers or mountains. Your presence and your words are a tremendous comfort to Us. We approve of what was said in your name, and We wholeheartedly accept your advice as being from a wise man who has seen and experienced much, 1 and We shall discuss it separately with you.

	To Margrave Albrecht of Brandenburg
	Benedini, 1755-1759
	Benedini, 1755-1759 MPL = Migne, Jacques-Paul: Patrologia latina. 217 vols. 1841-1865 [1]	3 vols. Lucca:
	• Collected Orations of Pope Pius II. Edited and translated by Michael v. Cotta-Schönberg. RTA = Deutsche Reichstagsakten
	12 vols. Copenhagen, 2019-2020 WO = Der Briefwechsel des Eneas Silvius Piccolomini. Hrsg. von Rudolf Wolkan. 3 vols. Wien, 1909-
	Voigt, Georg: Enea Silvio de' Piccolomini als Papst Pius der Zweite und sein Zeitalter. 3 vols. Berlin, 1918
	1856-63
	1 sed curasti : securitati J HA = Eneas Silvius Piccolomini: Historia Austrialis. Teil 1: Einleitung von Martin Wagendorfer. 1. 2 nostro desiderio : desiderio nostro G Redaktion ed. von Julia Knödler. Teil 2: 2./3, ed. Martin Wagendorfer. 2 vols. Hannover, 3 retineri vera caritas : vera charitas retineri G 2009. (Monumenta Germaniae Historica. Scriptores Rerum Germanicarum. Nova Series; 4 omit. J
	5 XLta G 24) 6 proximo anno Norimbergae : Norimbergae proximo anno G
	7 dissidii G, J
	8 cum G
	9 manibus G
	10 mirantur J

ecclesiastici ab anno MCXCVIII ubi Card. Baronius desinit. Auct. Odoricus Raynaldus. Tom. XVIII-XIX. Roma: Varesius, 1659-1663 2 Mansi, Giovanni: Ad concilia Veneto-Labbeana Supplementum. Sanctorum conciliorum et decretorum collectio nova: seu collectionis conciliorum Supplementum. vols. Luca: ex typographia Josephi salani, & Vincentii Junctinii, 1748-1752 Moore, John C.: Pope Innocent III (1160/61-1216) -To Root Up and to Plant. Leiden, 2003 Pastor, Ludwig: Geschichte der Päpste im Zeitalter der Renaissance. 16 vols. Freiburg i.B., 1886-1933 Pius II: Commentarii rerum memorabilium quae suis temporibus contigerunt. [1464] • Pius II: Commentarii rervm memorabilivm que svis temporibus contigervnt. Ed. A van Heck. II vols. Città del Vaticano: Biblioteca Apostolica Vaticana, 1984 (Studi e testi; 312-313) • Pius II: Commentaries. Ed. M. Meserve and M. Simonetta. Vols. 1 ff. Cambridge, MA: Harvard University Press, 2003 ff. Pius II: Orationes. [1436-1464] • Pius II: Orationes politicae et ecclesiasticae. Ed. Giovanni Domenico Mansi. Zimolo, Giulio C. (ed.): Le vite di Pio II di Giovanni Antonio Campano e Bartolomeo Platina. Bologna, 1964. (Rerum Italicarum Scriptores; t. III, p. II) 7. Sigla and abbreviations G = Lucca / Biblioteca Capitolare Feliniana / 544 H = Roma / Biblioteca Apostolic Vaticana / Chis. J.VII.251 J = Roma / Achivio Segreto Vaticano / Arm. XXXII 1 Abbreviations CO = Pius II: Commentarii rerum memorabilium quae suis temporibus contigerunt. [1464] HB = Aeneas Silvius Piccolomini: Historia Bohemica. Herausg. J. Hejnic & H. Rothe. 2 vols. Köln, 2005. (Bausteine zur slavischen Philologie und Kulturgeschichte. Neue Folge. Reihe B; 20) MA = Pius II: Decretum = Decretum magistri Gratiani. Ed. Lipsiensis secunda. Eds. A.L. Richter & A. Friedberg. 2 vols. Leipzig, 1879 Epistolarium = Enee Silvii Piccolominei Epistolarium Secvlare. Ed. A. van Heck. Città del Vaticano, 2007 Rainaldus = Annales ecclesiastici ab anno MCXCVIII ubi Card. Baronius desinit. Auct. Odoricus Raynaldus. Tom. XVIII-XIX. Roma: Varesius, 1659-1663 Reject = Reject Aeneas, accept Pius : Selected letters of Aeneas Sylvius Piccolomini (Pope Pius II). Intr. and transl. by T.M. Izbicki et al. Washington, D.C., 2006 Ad Albertum marchionem Brandeburgensem [

  Many are the places where you have raised memorials of victory, and you have enriched your temples with the spoils from your enemies. If there is anybody today whose fame will last, it is certainly you. An author would insult your name if he did not compare it to the generals of Antiquity, admired by Greece and Rome.[3] Verum quamvis te justa secutum bella putemus, inter Christianos tamen hactenus proeliatus es, quod difficile sine peccato fieri potest. Saepe innocens sanguis {182v} effunditur, interveniunt puellarum et nuptarum raptus. Dum superbit impius, incenditur pauper, diripiuntur ecclesiae, monasteria incenduntur, divina et humana miscentur. Apud victorem raro moderatio magis quam libido vindictae potest. Multa urgent conscientiam tuam, non dubitamus: non potuisti non laedere multos.[4] At haec expiare Turconico bello facile poteris. Pugnasti pro mundo hactenus; pugna nunc pro Christo. Pugnasti pro corpore; pugna nunc pro anima. Pugnasti pro fragili et caduca gloria; pugna nunc pro immortali et aeterna laude. Exerce te in hoc 1 bello, in quo possis omnes veteris culpae maculas expurgare, et te gratum creatori tuo reddere. {183v} Age, amplectere volenti animo hanc expeditionem, in qua si te armatum esse senserimus, magnam victoriae fiduciam capiemus. Nam te alterum in proelio David futurum speramus, de quo canebant, quia Saul mille et David X millia stravisset hostium. Tua persona pro magno nobis erit exercitu, et in te alterum Judam Macchabaeum, aut alterum Samsonem esse censebimus. Confortare igitur et esto robustus, et cum redieris ad fratres tuos, Germaniae proceres, stude, ut sedatis domesticis contentionibus, si quae pullulant, exercitus, qui promissus est, in Germania confletur, et in hostes ducatur. Reliqua dominus deus pro sua misericordia felicia esse praestabit. Amen.

Until now you have conducted many wars: in

Franconia, your fatherland; in Meissen, in Saxony, in Bohemia, in Silesia, in Poland, in Hungary, in Austria, in

Bavaria and in Swabia. There is no region in Germany where you have not sweated and toiled. You have been at war ever since your tender years 2 , first as a splendid solider, later as a commander. It is not without reason that some have called you the German Achilles.

3

Often you took spoils from the enemy. Often you were the first climb up the rampart and the enemy's walls.

"Non ullam" of Hertnidt vom Stein 1 to Pope Pius II (December 1459, Mantua)

  According to Matthias Thumser, Hertnidt vom Stein arrived at the Papal Court, then residing in Siena, probably in the beginning of March 1459, to present Margrave Albrecht Achilles' declaration of obedience to the new pope. Afterwards, he stayed with the court and followed the pope to Mantua. 2 He left Mantua before September 1459. Thumser does not mention him as a companion to Albrecht Achilles von Brandenburg, when this prince came to Mantua towards the end of December 1459: Nur kurze Zeit darauf und wahrscheinlich entgegen den Instruktionen seines Herrn verliess er Mantua und die Kurie und machte sich auf den Weg nach Bamberg … So musste der Fürstenkongress im September desselben Jahres seine Beratungen ohne ihn aufnehmen. … Obwohl Markgraf Albrecht um die Jahreswende doch noch persönlich nach Mantua kam un auch er feierlich seine Mitwirkung beim Kreuszug gegen die Türken beteuerte, blieben Taten letztlich aus.3 That Hertnidt vom Stein actually accompanied Albrecht Achilles on his journey to Mantua has been confirmed by Thumser in a personal communication of 30 June 2019.4

Hertardum de Lapide, decanum Parmbergensem, praesente illustri domino Alberto, Marchione Brandiburgensi, nomine et vice dignationis suae in ecclesia cathedrali in die epiphaniae immediate post missam celebratam per reverendum dominum cardinalem Rothomagensem [

  1] {52r} Non ullam, pontifex alme, huic principi meo negligentiam assignabis, si, ut summo opere exoptabat, tuam tam cito haud adierit sanctitatem. Non profecto Arpinati 1 illa explicari posset lingua, quanto desiderio quantaque gratia ac diligentia videre te et coram alloqui concupierit, cum praesertim se eum profiteatur, qui pro innata tibi humanitate et singulari virtute magis tibi quam sibimetipsi debere soleat. Nonnullorum enim, qui potius bello quam pace gaudent, impedimento detentus ab itinere hucusque ad te revocatus defecit. Suam tamen in te mirificam oboedientiam, devotionem et affectum singularem demonstrare cupiens, gubernandae 2 patriae causa postposita, uxoriae affectionis zelo semoto 3 ac cunctis suis, quae permaxima sunt, negotiis posthabitis, bello nunc paene fulminantem patriam deserere et tuae sacratissimae beatitudinis pedes promptus petere non dubitavit.[2] Ignosces igitur, alme pontifex, potissimum mihi, qui ista oratoria arte omnino nudus in medio adire haud dubitaverim. Non enim temeraria honoris ambitio aut major gloria quaedam 4 ad hoc me compulit, sed potius huis principis mei de me siquidem bene meriti officiorum magnitudo, cum equidem pro singularibus virtutibus suis et animi magnitudine obnoxium me etiam atque etiam profiteor. Porrecta igitur fronte accipies, accipies -inquam -barbaram hanc potius orationem quam luculentam. Non enim ex iisdem orta est locis, ubi tenuiori -ut ajunt -Minerva, sed potius pinguiori homines erudiri solent. Me enim abunde satis dixisse puto, sed oratorio isto dicendi genere derelicto hujus principis mei tantum denudavero tibi voluntatem, cum igitur, clementissime pontifex, qualem vides, offero.[3] Cujus animi magnitudo, corporis vires {52v} in rebus agendis, consilium in capiendis, administratio in gerendis, perseverantia ac labor haud umquam in eo, quod exoptaveras, tuae defuerint sanctitati. Non enim, ut nonnulli facere consueverunt, verborum prolixione homines sibi conciliare solet, sed quod semel dixerit, id regere vehementer exoptat.[4] Primus igitur tibi, alme pontifex, immensas suo nomine et ago et habeo gratias, qui pro tua singulari in eum pietate postquam negotiorum, quorum te gratia in hanc urbem recipere et Christiani nominis principes accersiri voluisti, salubrem deliberationem fecisti, ut ipsum iter jam agentem intellexisti, in hos dies operiri eum dignatus es. Verum etsi non tanti se meriti aut dignitatis esse cognoscat, adauctae tamen affectionis et paternae tuae in eum pietatis officia, quibus eum adhuc te in minoribus existente fovebas, mirum in modum jucundum reddidere. Exhortantur etiam eum et sibi animum adjungent accumulatissime tibi attributae virtutes non tibi solum, sed apostolicae sedi et universae Christianae religioni plurimum congratulari te hac nostra tempestate in summi apostolatus apicem deductum esse, cum Christianae fidei necessitati magna virtute, ingenti animo, plurimo labore succurendum sit atque prae ceteris in utramque partem exercitatus et tam pii operis percupidus omnium judicio comprobatus haberis.[5] Non sane latet principem hunc meum, quanto consiliorum pondere, quanta animi magnitudine, quanta prudentia ac fortitudine, cum te Germani tenerent regiones, reipublicae et Romani praesulatus auctoritatis ferventissimum amatorem, eruditissimum expertissimumque advocatum et procuratorem exhibueris. Hac consilia principum, hic regum conventus, hic urbium concursus ac potentatuum et nobilium plebeorumque exercitus aderant, quibus omnibus operae et sollicitudines tuae non deerant, propter quod primus in aula Caesareae majestatis venerabaris.[6] Fuit nonnumquam et hic princeps 1 majus consiliorum tuorum particeps. Saepenumero audivit te cum summa animi admiratione inflexibiles Germanorum animos singulari eloquentia exorantem in tuam sententiam duxisse. Et ut ex reliquis unum, quorum tamen princeps ipse plurima saepe solet ex innumeris virtutibus tuis recensere, commemorem quam divino eloquio in Francfordensi contione cum de expeditione contra inimicos Christi provide ageretur, eorum principum et legatorum, qui aderant, animos in commiserationem commoveras, ut neminem ipsorum hic audiveris, qui non se adjutorem hujus profectionus sanctae adversus Christianae gentis turbatores profitentur. Multa denique commemorare solet, quae tua prudentia consilio, auctoritate, cura, gratia et sollicitudine in catholicae fidei utilitatem ac sedis tuae auctoritatem et conservandam et augendam ingenii tui subtilitate persuadendi acumen obtinueris, dicendi faciendique sollertia fueris consecutus.[7] Jure igitur cum universa Christianorum plebe gaudendum sibi esse arbitratur eum pontificem, cunctorum moderatorem in terris et coadjutorem Dei creatum esse, qui priusquam dignitatem fuerit consecutus, eam prae cunctis tuebatur. Et veluti bonus imperator exercitus nonnisi virtute bellandi expertus {53r} et ex multis laudabiliter consecutis victoriis comprobatus 2 diligitur, sic te quoque, qui inter tantos vitae fluctuosas undique procellas et sirenarum cantus naufragae paene naviculae Petri succurrebas ejusque nonnumquam gubernatori ferens auxilium, temone appenso in solidum, adjutabas producere portum, non injuria uti vigilem et expertum nauclerum in summum gubernatorem delectum hac in parte salubriter Romanae sedi consultum esse vehementer exultat, quod ad hujus tuae profectionis in hanc urbem desideratam utilitas … 1 testatur.

Bibl. Apostolica Vaticana / Ottobon. Lat. 905,

Oration "Advenisse te citius"[53]

On Piccolomini's use of sources in general, see Collected orations of Enea Silvio Piccolomini / Pope Pius II, ch. 8.

References to the Annales are usually given in this form: (e.g.) Rainaldus, ad ann. 1459, nr. 67 (without reference to a specific edition). In the bibliographies of the individual orations, the first edition is usually listed, but the Theiner edition is actually the preferred one since it is the easiest available on the web

Cf. Homer: Odyssey, 1.1

"unguiculis"

Indeed, Pius himself was the first to call him so

in hoc : hoc in J

Innocentius III: Registrum, 11:

(MPL 215: 1545): Pugnastis fortassis hactenus pro gloria transitoria, pugnate iam pro gloria sempiterna. Pugnastis pro corpore, pugnate pro anima. Pugnastis pro mundo, pugnate pro Deo 2 1.Kings, 18, 7

Herdtnidt vom Stein (c. 1427 -1491): Chancellor of Margrave Albrecht Achilles of Brandenburg, and later Dean at the Cathedral of Bamberg. See Thumser. Various latinized forms of his name are documented in contemporary sources, Hertnidus (Thumser, p. 7), Hartungus(Thumser, p, 10), H[e]rtinidus(Thumser, p. 15), Herthnidus(Thumser, p. 18), Hernildus(Thumser, p.

20), Hertindum(Longolius, p. 93)

Herdtnidt vom Stein (c. 1427 -1491): Chancellor of Margrave Albrecht Achilles of Brandenburg, and later Dean at the Cathedral of Bamberg. See Thumser. Various latinized forms of his name are documented in contemporary sources, Hertnidus (Thumser, p. 7), Hartungus(Thumser, p, 10), H[e]rtinidus(Thumser, p. 15), Herthnidus(Thumser, p. 18), Hernildus(Thumser, p.

20), Hertindum (Longolius, p. 93) 2 Thumser, pp. 36 ff. 3 Thumser, pp. 41-42 4 Man muss sich den zeitlichen Ablauf nun wohl folgendermaßen vorstellen: Hertnidt brach nach dem 23. Juli 1459 von Mantua nach Bamberg auf, sicherte sich dort seine päpstliche Provision und erhielt am 16. August Kanonikat und Dekanat. Das war eine flotte Reise, erscheint aber durchaus möglich. Einige Monate später kehrte er, nun zusammen mit dem Markgrafen, nach Mantua zurück und hielt dort die Rede. Die Annahme, dass er zwischendurch in Bamberg war, stützt seine Bezeichnung als Dekan in der Überschrift zur Rede.

I.e. Cicero, who was born in Arpinum

gubernande cod.

scemoto cod.

quendam cod.

Albrecht von Brandenburg

comprobatis cod.

scemotis cod.

continue cod.

tantum add. cod

26

detentus fuit, ut tua potuit clementia jamdudum novisse partimque ab eo semotis 1 arbitris intelliget tibi morem gererere, etsi in eam deliberationem incidit, quam salubriter conclusam intelligit, tametsi nihil consultatio sua praeter divina illa ingenia, quae ex tua diligentia ad eam rem et exquisita et audita sunt, afferre posset. Nam quod suae fuerit sententiae, novit sanctitas tua, ipsum in Franckfordensi conventu 2 , cum simul imperiales collegae fuere, edidisse juxta 3 facultatem suam offerre se ad tam sanctam expeditionem praesto esse. Et futuris non deerit diebus legatis tuis in Nurembergensi dieta secundum facultatem personae et dominorum suorum et cooperari et efficere, quod tibi placitum et Christianae religioni utile videbitur.

Appendix: Oration "Non ullam" of Hertnidt vom Stein 1 to Pope Pius II (6 January 1460, Mantua)