A Manifold Learning Approach to Data-Driven Computational Elasticity and Inelasticity

Arch Computat Methods Eng DOI 10.1007/s11831-016-9197-9

S.I.: MACHINE LEARNING IN COMPUTATIONAL MECHANICS

1 1 1 Rube´n Iban˜ez • Emmanuelle Abisset-Chavanne • Jose Vicente Aguado • 2 2 1 David Gonzalez • Elias Cueto • Francisco Chinesta

Received: 4 October 2016 / Accepted: 13 October 2016 Ó CIMNE, Barcelona, Spain 2016

Abstract Standard simulation in classical mechanics is explicit, often phenomenological, models. This technique based on the use of two very different types of equations. is based on the use of manifold learning methodologies, The first one, of axiomatic character, is related to balance that allow to extract the relevant information from large laws (momentum, mass, energy,...), whereas the second experimental datasets. one consists of models that scientists have extracted from collected, natural or synthetic data. Even if one can be confident on the first type of equations, the second one 1 Introduction contains modeling errors. Moreover, this second type of equations remains too particular and often fails in Big Data has bursted in our lives in many aspects, ranging describing new experimental results. The vast majority of from e-commerce to social sciences, mobile communica- existing models lack of generality, and therefore must be tions, healthcare [16], etc. However, very little has been constantly adapted or enriched to describe new experi- done in the field of scientific computing, despite some very mental findings. In this work we propose a new method, promising first attempts. Engineering sciences, however, able to directly link data to computers in order to perform and particularly Integrated Computational Materials Engi- numerical simulations. These simulations will employ neering (ICME) [12], seem to be a natural field of axiomatic, universal laws while minimizing the need of application. In the past, models were more abundant than data, too expensive to be collected and analyzed at that time. & Francisco Chinesta However, nowadays, the situation is radically different, [email protected] data is much more abundant (and accurate) than existing Rubeń Iban˜ez models, and a new paradigm is emerging in engineering [email protected] sciences and technology. For instance, high-energy physics Emmanuelle Abisset-Chavanne experiments produce some 1Pb of data per day, while in [email protected] 2012, 162,000 papers were published in materials science Jose Vicente Aguado and engineering journals. [email protected] Advanced clustering techniques, for instance, not only David Gonzalez help engineers and analysts, they become crucial in many [email protected] areas where models, approximation bases, parameters, etc. Elias Cueto are adapted depending on the local state (in space and time [email protected] senses) of the system [1, 9]. They make possible to define hierarchical and goal-oriented modeling. Machine learning 1 High Performance Computing Institute and ESI GROUP Chair, Ecole Centrale de Nantes, 1 Rue de la Noe, [8] needs frequently to extract the manifold structure in 44300 Nantes, France which the solution of complex and coupled engineering 2 Aragon Institute of Engineering Research, Universidad de problems is living. Thus, uncorrelated parameters can be Zaragoza, Zaragoza, Spain efficiently extracted from the collected data, coming from 123 R. Iban˜ez et al. Z Z numerical simulations or experiments. As soon as uncor- ruÃ : r dx ¼ uÃ Áðr Á nÞ dx; related parameters are identified (constituting the infor- X C mation level), the solution of the problem can be predicted at new locations of the parametric space, by employing where n represents the outward unit vector normal to the adequate interpolation schemes [5, 10]. On a different boundary. setting, parametric solutions can be obtained within an If we consider C ¼ CD [ CN ,(CD \ CN ¼;), repre- adequate framework able to circumvent the curse of senting portions of the domain boundary where, respec- dimensionality for any value of the uncorrelated model tively, displacements u ¼ ugðxÞ (Dirichlet boundary parameters [4]. conditions) and tractions r Á n ¼ tgðxÞ (Neumann boundary This unprecedented possibility of directly determine conditions) are enforced, the weak form finally reads: 3 knowledge from data or, in other words, to extract models Find the displacement field u 2ðH1ðXÞÞ satisfying the from experiments in a automated way, is being followed essential boundary conditions uðx 2 CDÞ¼ugðxÞ such that Z Z with great interest in many fields of science and engi- Ã Ã neering. For instance, the possibility of fitting the data to a e : r dx ¼ u Á t dx; ð1Þ X CN particular set of models has been explore recently in [2]. Ã Willcox and coworkers, on the contrary, have established a 8u regular enough and vanishing on CD, i.e. strategy that allows to construct reduced-order models Ã 1 3 8u 2H0ðXÞÞ . from data, by inferring the full-order operators without the In the previous weak form, the symmetry of r implies need to construct them explicitly, nor having a direct the equality ru : r ¼rSu : r, with rSu the symmetric knowledge on the governing models [13, 14]. Closely component of the displacement gradient, also known as related, Ortiz has developed a method that works without strain tensor, generally denoted by e. constitutive models, by finding iteratively the experimental The weak form given by Eq. (1) involves kinematic and datum that best satisfies conservation laws [6]. dynamic variables from the test displacement field uÃ and the In the ICME framework of materials modeling, design, stress tensor r respectively. In order to solve it a relationship simulation, and manufacturing, this subtle circle is closed linking kinematic and dynamic variables is required, the so- by linking data to information, information to knowledge called constitutive equation. The simplest one, giving rise to and finally knowledge to real time decision-making, linear elasticity, is known as Hooke’s law (even if, more than opening unprecedented possibilities within the so-called a law, it is simply a model), and writes DDDAS (Dynamic Data Driven Application Systems) [3, 11]. r ¼ kTrðeÞI þ le; ð2Þ In the present work we will assume that all the needed where TrðÁÞ denotes the trace operator, and k and l are the data is available. We will not address all the difficulties Lame coefficients directly related to the Young modulus E related to data generation or obtention from adequate and the Poisson coefficient m. experiments. This is a topic that, of course, remains open. By introducing the constitutive model, Eq. (2), into the On the contrary, we develop a method in which this stream weak form of the balance of momentum, Eq. (1), a problem of data plays the role of a constitutive equation, without the is obtained that can be formulated entirely in terms of the need of a phenomenological fitting to a prescribed model. displacement field u. By discretizing it, using standard To better understand the data-driven rationale addressed finite element approximations, for instance, and performing in the present paper, let us consider, for the sake of clarity, numerically the integrals involved in Eq. (1), we finally a very simple problem: linear elasticity. In that case the obtain a linear algebraic system of equations, from which balance of (linear and angular) momentum leads to the the nodal displacements can be obtained. existence of a symmetric second-order tensor r (the so- In the case of linear elasticity there is no room for dis- called Cauchy’s stress tensor) verifying equilibrium, cussion: the approach is simple, efficient and has been expressed in the absence of body forces, as applied successfully to many problems of interest. Today, rÁr ¼ 0: there are numerous commercial codes making use of this mechanical behavior and nobody doubts about its perti- The finite-element solution of this equilibrium equation nence in engineering practice. However, there are other starts from establishing a weak form in the domain X with material behaviors for whom simple models fail to describe boundary C oX, Z any experimental finding. These models lack of generality uÃ ÁrÁðÞr dx ¼ 0: (universality) and for this reason a mechanical system is X usually associated to different models that are progres- By integrating by parts, it results sively adapted and/or enriched from the collected data.

123 A Manifold Learning Approach to Data-Driven Computational Elasticity and Inelasticity

The biggest challenge could then be formulated as fol- on a curve (one-dimensional manifold) or on a surface lows: can simulation proceed directly from data by cir- (two-dimensional manifold). However, when dealing with cumventing the necessity of establishing a constitutive high dimensional spaces, a simple visual observation is, in model? In the case of linear elasticity it is obvious that such general, not possible. Moreover, the extraction of uncor- an approach lacks of interest. However, in other branches related features (often referred to as latent parameters) of engineering science and technology it should be an seems to be more physically pertinent. appealing alternative to standard constitutive model-based Therefore, appropriate manifold learning (or non-linear simulations. In our opinion, we are at the beginning of a dimensionality reduction) techniques are needed to extract new era, the one of data-based or, more properly, data- the underlying manifold (when it exists) in multidimen- driven engineering science and technology, where as much sional phase spaces. A panoply of techniques exist to this as possible data should be collected and information end. The interested reader can refer to [1, 15, 17–19], just extracted in a systematic way by using adequate machine to cite a few references. In this work we focus on the learning strategies. Then, simulations could proceed particular choice of Locally Linear Embedding—-LLE— directly from this automatically acquired knowledge. techniques [17]. This method proceeds in two steps: Thus, the question from a methodological viewpoint 1. Each point X , m ¼ 1; ...; M is linearly interpolated could be reformulated as: If Hooke had never existed, linear m from its K nearest neighbors. In principle K should be elasticity ﬁnite element simulations would have existed? greater that the expected dimension d of the underlying This paper addresses this question, trying to push it manifold and the neighbors should be close enough so beyond linear elastic behaviors. Next section focuses on the as to ensure the validity of linear approximation. In construction of the so-called constitutive manifold from the general, a small but enough number of neighbors K and collected data. Section 3 deﬁned the manifold-based data- a large-enough sampling M ensures a satisfactory driven framework, and Sect. 4 introduces data-driven reconstruction. For each point X we can write the simulation in the context of elastic models (linear and m locally linear data reconstruction as: nonlinear). Finally, Sect. 5 extends the procedure to X inelastic behaviors. Xm ¼ WmiXi; ð3Þ

i2Sm where W are the unknown weights and S the set of 2 Collecting Data and Constructing mi m the K-nearest neighbors of Xm. the Constitutive Manifold If we perform this locally linear interpolation for every data point in the high dimensional phase space, the set Imagine, to begin with (more general scenarios will soon of weights that best approximates the manifold struc- be considered) mechanical tests conducted on a perfectly ture of the data will be obtained by minimizing the linear elastic material, in a specimen exhibiting uniform functional stresses and strains. As previously indicated, in this paper 2 we do not address issues related to data generation. Thus, XM XM W X W X ; for M randomly applied external loads, we assume our- Fð Þ¼ m À mi i m¼1 i¼1 selves able to collect M couples ðrm; emÞ, m ¼ 1; ...; M. These pairs could be represented as a single point Xm in a where Wmi is zero if Xi does not belong to the set of K- phase space of dimension D ¼ 12 (the six distinct com- nearest neighbors of Xm. ponents of the stress and strain tensors, respectively). In the 2. We assume now that each linear patch around Xm, 8m, sequel Voigt notion will be considered, i.e. stress and strain is mapped onto a lower dimensional embedding space tensors will be represented as vectors and the fourth-order of dimension d D. To maintain the neighborhood elastic tensor reduces to a square matrix. structure of the set (other methods like isomap [19] Each vector Xm thus deﬁnes a point in a space of conserve distance in the embedding space instead), dimension D and, therefore, the whole set of samples weights are assumed to remain unchanged in the low- represents a set of M points in RD. We conjecture that all dimensional, embedding space. The problem thus these points belong to a certain low-dimensional manifold becomes the determination of the coordinates of each embedded in the high-dimensional space RD. Imagine for a point Xm in the low dimensional embedding space, d while that the M points belong to a curve, a surface or a nm 2 R . hyper-surface of dimension d D. When D ¼ 3 a simple For this purpose a new functional G is introduced, that observation sufﬁces for checking if these points are located depends on the searched coordinates n1; ...; nM

123 R. Iban˜ez et al.

2 depicts the location of samples n ¼ nðX Þ¼nðr ; e Þ XM XM m m m m into the resulting two-dimensional manifold, as well as the Gðn1; ...; nMÞ¼ nm À Wmini ; m¼1 i¼1 associated elastic energy of each sample, showing that LLE where now the weights are known and the reduced preserves the smoothness of the elastic energy field of the sample in the embedding space. coordinates nm are unknown. The minimization of functional G results in a M Â M eigenvalue problem whose d-bottom non-zero eigenvalues define the set of orthogonal coordinates in which the manifold is map- 3 Working with Constitutive Manifolds ped. We have abandoned the idea of a phenomenological con- It is important to note that functional Gðn ; ...; n Þ, 1 M stitutive equation. Instead, we have defined the concept of with the different coordinates nm already calculated as just described, offers an error estimator on the locally (experimentally obtained) constitutive manifold, as the one linear embedding capacity, and even a local estimator with a minimal number of latent parameters (embedding coordinates) in which the state of the sample will evolve in can be derived by considering different stress and strain conditions. XM However, for the method to be useful, we need to define Eðn Þ¼ n À W n : ð4Þ m m mi i a strategy to solve problems stated in weak form and dis- i¼1 cretized by finite elements. Several options can be considered, which are described next. Thus, if we consider the introduction of a new point n in the embedding space Rd after identifying its neighbors set 1. Identifying the locally linear behavior. If we consider SðnÞ and calculating the locally linear approximation locally linear approximations, fully justified if EðnmÞ, weights, we can come back to RD and reconstruct X from given by Eq. (4), remains small enough at each its neighbors Xi, i 2SðnÞ. position nm (if it is not the case the sampling should be In the linear elastic behavior the application of the just improved locally or globally), we can write described technique results, as expected, in a flat manifold XM of dimension two, i.e. d ¼ 2. This is in perfect agreement nm ¼ Wmini; to the fact that Hooke’s law is completely characterized by i¼1 two coefficients (either Young’s modulus and Poisson with Wmi ¼ 0ifi 62Sm and where nm is a stress–strain coefficient, or Lame’s coefficients) and is linear. Figure 1 couple. This implies a locally linear elastic behavior, that allows obtaining the elastic tensor C from Xm and Xi (related to nm and ni respectively), with i 2Sm,by minimizing the functional × 6 X 10 2 3 HðCÞ¼ ðri À C Á eiÞ :

i2Sm 1 2.5 This results in the obtention of CðXmÞCm.

0.5 2. Identifying the locally linear tangent behavior. In order 2 to consider Newton strategies the locally tangent linear 0 behavior should be computed. Again, it is easy to obtain by considering Dmi Xm À Xi ¼ðrm À -0.5 1.5 m m ri; em À eiÞ or Dmi ¼ðDri ; Dei Þ, i 2Sm. Because of the locally linear behavior around point Xm, we can -1 1 write 4 m m 2 Dri ¼ CT Á Dei ; ð5Þ × 12 10 0.5 0 3 that allows deﬁning the functional HT ðCT Þ 2 X ÀÁ -2 × 1013 m m 2 1 HT ðCT Þ¼ Dri À CT Á Dei ; ð6Þ -4 0 i2Sm

Fig. 1 Reduced coordinates nm on the resulting two-dimensional whose minimization results in the tangent elastic ten- manifold. The color map represents the associated elastic energy. sor CT ðXmÞCT;m. (Color ﬁgure online)

123 A Manifold Learning Approach to Data-Driven Computational Elasticity and Inelasticity

3. No model at all. The third level of description that, once introduced into the weak form, reads Z considers points Xm without trying to identify local Ã behavior models at all. e ðxÞÁðÞCT ðxÞÁDeðxÞ dx X Z Z It is important to note that even if the just discussed ¼À eÃðxÞÁðÞCðxÞÁeðxÞ dx þ uÃðxÞÁtðxÞ dx: descriptions are based on the original manifold Xm and not X CN on the reduced one nm, the consideration of the reduced manifold allows to obtain a global view of the manifold 3. If no local behavior has been identified, the only dimensionality as well as safer interpolations on the man- knowledge consists of the experimental data. In these ifold. This ensures that interpolated data n belongs to the circumstances we propose to consider a mixed formu- manifold, before applying the inverse mapping to obtain X lation involving the two unknown fields eðuÞ and r as on the original manifold. considered in the LaTIn method [8]. We consider a simple solution strategy consisting on an iteration between two manifolds, the first one related to (e,r) 4 Data-Driven Simulation in the Elastic Case couples verifying equilibrium Eq. (1); and the second one related to couples (e^; r^) verifying the (unknown) We assume that the elastic behavior is accessible from the constitutive equation—in other words, belonging to the data contained into the so-called constitutive manifold but constitutive manifold. The iteration solver sketched in that an explicit expression relating stresses and strains is Fig. 2, depicts the usually non linear constitutive neither available nor desired. Immediately, a question manifold (red curve) and the equilibrium one (blue arises on how to solve the weak form related to the equi- straight line). The problem solution is found at the librium of the mechanical system given by Eq. (1)ifno intersection of both manifolds. closed-form expression on r ¼ rðeÞ is available. If we assume that, at iteration n, the couple ðen; rnÞ verifies the equilibrium, and that it does not belong to 4.1 Discretization Schemes the constitutive manifold, a new couple ðe^; r^Þ is sought by considering an appropriate search direction from In this case we could consider three different approaches ðen; rnÞ. In fact the searched couple is no more that the depending on the chosen behavior description as just dis- intersection of the search direction with the constitu- cussed in the previous section: tive manifold. The just updated stress–strain couple belongs to the constitutive manifold, but it does not 1. From the just identified locally linear behavior CðXÞ verify equilibrium. Thus, a new equilibrated solution one could apply the simplest explicit linearization enþ1; rnþ1 is searched from the former one, being the technique operating on the standard weak form ð Þ Z Z intersection of a new search direction and the equilib- eÃðxÞ : rnþ1ðxÞ dx ¼ uÃðxÞÁtðxÞ dx; ð7Þ rium manifold. The iteration process continues until X CN reaching the problem solution at the intersection of where at each point, from the stress–strain couple at both manifolds. position x, XðxÞ, the locally linear behavior CðXðxÞÞ The just described procedure requires a local step for can be obtained (in practice at the Gauss points used the computation of the couple ðe^; r^Þ at each integration for the integration of the weak form) that allows us to point considered in the weak form, Eq.(1), and a global write (using Voigt notation) Z Z eÃðxÞÁðÞCðxÞÁeðxÞ dx ¼ uÃðxÞÁtðxÞ dx: X CN This allows, in turn, to compute the displacement field and from it, to update the strain and stress fields, to compute again the locally linear behavior. The process continues until convergence. 2. From the just identified locally linear tangent behavior CT ðXÞ one could apply a Newton linearization tech- Fig. 2 A generic nonlinear iteration solver between the constitutive nique where manifold (red curve) and the equilibrium manifold (blue straight or line), representing the locus of the points satisfying the weak form of rðe þ DeÞ¼rðeÞþ De ¼ rðeÞþC Á De; the problem in mixed form, Eq. (7). (Color figure online) oe T

123 R. Iban˜ez et al.

step in which the weak form is solved with the unknown. Thus, our strategy is composed of a behavior known at all the integration points. In what sequence of nonlinear-local and nonlinear-global follows we describe both steps. problems, trying to avoid a priori choices of D. Obviously if the last is fixed, global problems – Local step become linear as it is the case when considering the At each integration point x , g ¼ 1; ...; ngp,we g LaTIn linearization technique. Moreover, the dis- consider ðenðx Þ; rnðx ÞÞ and look for g g crete global matrix does not change during the ðe^ðx Þ; r^ðx ÞÞ. Even if there is an infinity of g g iterations. However, we would like to emphasize possible search directions, a natural choice consists that our objective is to solve a constitutive model- in projecting it onto the constitutive manifold. free problem, more than addressing nonlinear – Global step issues. From the strain-stress couples satisfying the con- Thus, we distinguish two type of iterations, the so- stitutive law at every integration point, we come called global-local ones that involves the determi- back to the weak form, Eq. (1), in order to obtain nation of stress–strain couples verifying the consti- updated strain-stress couples satisfying equilibrium tutive equation and then their updating to ensure nþ1 nþ1 ðe ðxÞ; r ðxÞÞ, x 2 X. equilibrium (as illustrated in Fig. 2). Then a second The generic search direction can be written as: iteration is needed for solving the nonlinear global rnþ1ðxÞÀr^ðxÞ¼D Áðenþ1ðxÞÀe^ðxÞÞ; ð8Þ problem in order to compute the stress–strain couple verifying equilibrium when the searching direction with D a symmetric positive-definite matrix to D is assumed unknown. This induces an additional ensure the problem ellipticity discussed below. nonlinearity in the global equilibrium problem. Enforcing now the equilibrium Z Z At this point two possibilities exist: eÃðxÞÁrnþ1ðxÞ dx ¼ uÃðxÞÁtðxÞ dx; a. Considering a single direction D, the same for X CN every Gauss point for which the behavior was and using Eq. (8), it results determined. Each of them is represented by a Z ÀÁ point on the constitutive manifold. In that case Ã nþ1 e ðxÞÁ r^ðxÞþD Áðe ðxÞÀe^ðxÞÞ dx in order to determine the stress–strain couple Z X satisfying equilibrium as well as the optimal Ã ¼ u ðxÞÁtðxÞ dx; direction D, we are enforcing Eq. (9) as well as C N the fact that the searched couple that can be rewritten as nþ1 nþ1 Z Z ðe ðxÞ; r ðxÞÞ must be the closest point ÀÁ to the constitutive manifold. This optimality eÃðxÞÁ D Á enþ1ðxÞ dx ¼À eÃðxÞ X X condition writes Z ÀÁ 2 Á ðÞr^ðxÞÀD Á e^ðxÞ dx þ uÃðxÞÁtðxÞ dx: D ¼ arg min rnþ1ðx; DÃÞÀr^Ã DÃ CN ÀÁ ð10Þ 2 ð9Þ þ enþ1ðx; DÃÞÀe^Ã ; Matrix D should provide the fastest convergence where ðr^Ã; e^ÃÞ is the closest point on the con- rate while ensuring the problem ellipticity. To stitutive manifold to the stress–strain couple 2 ensure its positivity we can consider D ¼ B with related to the direction DÃ. T B symmetric, i.e. B ¼ B, and look for B instead of Obviously the solution requires some iterations D. to reach the minimum distance that will be in The a priori choice of direction D is not obvious in general (except when considering linear most of problems. In the case of the LaTIn method behaviors) non-zero because we consider the [8] this matrix is assumed given when solving the same matrix D for all the Gauss points involved global problems precisely because it was proposed in the integration of the weak form (9). as a nonlinear solver able to decouple the local and b. We consider a field DðxÞ, that implies the nonlinear problem from the global but linear one. increase of the number of degrees of freedom. In our case, we are considering a mixed formula- However, by considering for example a differ- tion for solving a problem without an explicit ent matrix at each Gauss point, the minimiza- knowledge of the constitutive equation. The most tion problem given by Eq. (10) leads to the general option consists of considering matrix D

123 A Manifold Learning Approach to Data-Driven Computational Elasticity and Inelasticity

problem solution in a single iteration. The constitutive manifold. In a general setting, this manifold employ of a coarse mesh to approximate D is a should come from experiments, but in this case was gen- nice compromise between the two limit cases: erated in silico by assuming a linear elastic behavior with considering a single search direction or one at an unit elastic modulus. each Gauss point. The use of strategies based on the identification of the locally linear behavior or its tangent counterpart allows as expected (due to its linear behavior) solving the problem in 4.2 A First Numerical Example: A Beam Subjected a single iteration. It is important to note that both strategies to Simple Traction are weakly intrusive, making possible its implementation into any commercial simulation code with the only dif- In order to illustrate the data-driven procedure, we consider ference that the updated locally linear behavior comes form first a linear elastic beam subjected to simple traction and a data table instead of any mathematical expression. solve the associated 1D equilibrium problem. Different In what follows we are discussing the use of the third scenarios are considered and discussed below. strategy. The equilibrium manifold and the different strain- First, the beam is assumed clamped at its left boundary stress couples at the different iterations are depicted in x ¼ 0 with a constant unit force F ¼ 1 applied at its right Fig. 3 for D ¼ 10, D ¼ 2 and D ¼ 1. These D-values boundary x ¼ 1. Because of the expected simple solution represent in fact different search directions in Fig. 2. It can only 5 linear finite elements were considered for dis- be noticed that when D ¼ 1 is chosen, this value coincides cretizing its equilibrium weak form. Figure 3 depicts the with the elastic modulus associated to the constitutive manifold, and therefore convergence is reached in a single iteration. All the simulations started by assuming the same stress–strain couple ðr0; e0Þ¼ð3:0; 3:0Þ at every Gauss point. In these figures, the search direction in the global problem D was fixed ‘‘a priori’’. When the strategy described in the previous section is used, implying the determination of the optimal value of D, the nonlinear problem involving r, e and D, with ðr0 ¼ 3; e0 ¼ 3; D0 ¼ 3Þ, converges in a single iteration of the local-global problem. This is so even if a few iterations were required for solving the nonlinear global problem, to obtain the reference values defining the problem solution ðr ¼ 1:0; e ¼ 1:0; D ¼ 1Þ. Because of the linearity of the constitutive manifold, no difference exists between considering a single direction D or a different one at each Gauss point. The solution is again obtained in a single global-local iteration and a few ones for solving the nonlinear global problem. In order to make the problem a bit more complex, we consider the previous one but now we consider an uniformly distributed traction along the beam length. Thus a linear stress and strain distribution is expected. In other words, each Gauss point will be at a state located at different points of the constitutive manifold. Figure 4 represents the stress–strain manifold along the beam length, where the stress–strain couples at the Gauss points are shown. It can be seen that when starting from the initial guess ðr0ðxÞ¼3; e0ðxÞ¼3; D0 ¼ 3Þ and again because of the linearity of the constitutive manifold, the convergence is reached in a single global-local iteration with few iterations for the solution of the nonlinear global Fig. 3 Beam subjected to traction: (top) D ¼ 10, (center) D ¼ 2 and problem. (bottom) D ¼ 1 123 R. Iban˜ez et al.

Fig. 6 Beam subjected to a traction for a nonlinear behavior: manifold-based Newton linearization

Fig. 4 Beam subjected to uniformly distributed traction

Finally, we consider a nonlinear constitutive law defined from points with a prescribed stress–strain relationship r ¼ E2, with E ¼ 1. In the case of a unit traction at the right boundary and when considering uniform initial strain and stress guesses on the constitutive manifold, all the Gauss points will have an identical behavior. When applying the fixed point linearization based on the locally linear manifold C or the Newton strategy making use of the locally linear tangent manifold CT , the procedure proposed in the previous section converges very fast. Iterations to convergence are depicted in Figs. 5 and 6 respectively. If, on the contrary, we proceed following the third strategy mentioned previously, i.e., directly from data, Fig. 7 Beam subjected to traction for a nonlinear behavior Fig. 7 depicts the initial guess and the solution after convergence ðrðxÞ¼1;ðxÞ¼1Þ. Here, D is unique and calculated at each global-local iteration. Moreover, at each Manifold-based locally linear behaviors resulting in the one of these iterations a nonlinear global problem must be fixed point and Newton strategies proceed faster that the solved needing for few extra-iterations. one based on the solution from the only knowledge of data. If we combine behavior nonlinearities and nonuniform However, it requires the identification of such behaviors solutions (e.g., a distributed traction along the bar) we with the subsequent errors that they could imply if coarse proved that the convergence can be improved by consid- samplings of the constitutive behavior are employed. ering a different D at each Gauss point with respect to the use of a single search direction D for all them, even if the 4.3 A Two-Dimensional Case Study global problem size increases significantly. We considered a 2D problem defined on a square involving again an elastic behavior defined from a manifold in the space ðr; eÞ. This constitutive manifold was proved to project onto a just two-dimensional one in its reduced form, as discussed previously. The square is clamped on its left boundary, free on the top and bottom sides and a unit traction is applied on its right side. Any of the proposed strategies, the ones making use of the manifold-based locally linear behaviors or the one proceeding directly from data, allow reaching the same converged solution depicted in Fig. 8. The last one employs a single search direction D or a different one at each Gauss Fig. 5 Beam subjected to a traction for a nonlinear behavior: manifold-based fixed point linearization point DðxÞ. It agrees in minute with the one obtained by 123 A Manifold Learning Approach to Data-Driven Computational Elasticity and Inelasticity

to the stress-elastic strain manifold) whereas the plastic contribution that involves the yield surface f ðrÞ is assumed given by its own manifold. Using again Voigt notation, the elastic behavior expressed from r ¼ C Á ee, where C represents the manifold-based elastic tensor and ee refers to the elastic component of the deformation (the reversible one). The total strain can be decomposed in its elastic and inelastic components, e ¼ ee þ ep; where we assume the plastic flow rate of ðrÞ e_p ¼ k ¼ kn; or where the yield surface f ðrÞ is provided by experimental data. To generate these data in silico, we assume that it follows a von Mises model f ðrÞ¼re À Y, with Y the yield stress (no hardening is considered) and re the equivalent stress related to the von Mises criterion. f ðrÞ results in the surface represented in Fig. 9 where, for the sake of clarity, it is represented in the space of stresses. The persistency condition f_ðrÞ¼0 when plastic flow occurs, results in the following plastic flow nT Á C Á e_ k ¼ ; nT Á C Á n or in its incremental counterpart nT Á C Á De k ¼ ; nT Á C Á n Fig. 8 2D problem associated to a ‘‘hidden’’ linear elastic behavior: (top) horizontal component of the displacement and (bottom) vertical with now Dep ¼ kn. component Here three fields must be considered, stress, strain and using standard model-based discretization. Again, a New- plastic strain. As soon as the last one is known, the elastic ton technique remains superior to the other choices. strain can be locally determined and the stresses obtained In what respects the solution accuracy there are different from the elastic manifold using the couple stress-elastic aspects affecting it: (1) the constitutive manifold sampling component of the strain. when nonlinear behaviors are addressed; (2) the finite In these expressions everything is properly defined element approximation and finally (3) the threshold con- except n, since we assume that the explicit form of the sider in the nonlinear iteration schemes. Even if a detailed yield condition, i.e. f ðrÞ is unknown and the only available analysis of the accuracy and rate of convergence is beyond data is the manifold depicted in Fig. 9. However, n is easily the aim of the present work, our numerical experiments indicate that convergence is assured by using fine enough samplings of the constitutive manifolds as well as by considering fine enough finite element discretizations.

5 Addressing Inelastic Behaviors: Linear Elastic- Perfectly Plastic Behavior

In this section we start by addressing the case of a linear- elastic-perfectly plastic 2D behavior. We assume the linear e e elastic contribution defined locally from CðX Þ (X refers Fig. 9 Plastic manifold associated to the von Mises plasticity case 123 R. Iban˜ez et al. accessible by considering the normal vector to the plastic stress space is depicted in Fig. 10. It can be noticed that the manifold depicted in Fig. 9. elastic behavior applies when the stress remains inside the Now one could imagine performing a standard linear plastic surface and then it remains in the surface during the elastic-perfectly plastic simulation by using a finite element plastic flow. Again, for the sake of simplicity, the results explicit code where the plastic deformation is computed are shown in the stress domain. Finally, Fig. 11 depicts the from the manifold that allows extracting n instead of the three components of the plastic strain for three different knowledge of function f ðrÞ and its explicit derivative with levels of the applied load acting on the right side of the respect to the stresses. clamped square previously considered. The different When considering the traction of a square domain along strategies allows to compute the same results.The Newton its right side, with appropriate boundary conditions on its algorithm results again to be the one involving less com- left side (with tension-free conditions on the top and bot- putational effort. tom boundaries) ensuring an homogeneous stress and strain Even if this analysis proved that we could proceed as fields everywhere in the domain, the stress trajectory in the usually when function f ðrÞ is not explicitly known, the elastic behavior was assumed given by the locally-linear elastic manifold. Obviously the extension to implicit for- mulations or to more complex nonlinear elastic behaviors again based on a locally-linear tangent description is straightforward.

6 Conclusions

This paper constitutes a ﬁrst attempt to reduce the model- Fig. 10 Stress trajectory in the stress space in the elastic-perfecly ing needs in computational mechanics. We proved that by plastic behavior knowing the different stress–strain couples deﬁning the

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