Arch Computat Methods Eng DOI 10.1007/s11831-016-9197-9 S.I.: MACHINE LEARNING IN COMPUTATIONAL MECHANICS A Manifold Learning Approach to Data-Driven Computational Elasticity and Inelasticity 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 Rube´n 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- ruà : r dx ¼ uà 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- à à 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]. à 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 à 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 uà 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 uà 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].