Data-Driven simulation of inelastic materials using structured data sets, tangent space information and transition rules
DData-Driven simulation of inelastic materials using structured datasets, tangent space information and transition rules
K. Ciftci , K. Hackl ∗ Institute of Mechanics of Materials, Ruhr-University Bochum, D-44801 Bochum, Germany.
Abstract
Data-driven computational mechanics replaces phenomenological constitutive functions byperforming numerical simulations based on data sets of representative samples in stress-strain space. The distance of modeling values, e.g. stresses and strains in integration pointsof a finite element calculation, from the data set is minimized with respect to an appro-priate metric, subject to equilibrium and compatibility constraints, see [1, 2, 3]. Althoughthis method operates well for non-linear elastic problems, there are challenges dealing withhistory-dependent materials, since one and the same point in stress-strain space might corre-spond to different material behavior. In [4], this issue is treated by including local historiesinto the data set. However, there is still the necessity to include models for the evolution ofspecific internal variables. Thus, a mixed formulation is obtained consisting of a combinationof classical and data-driven modeling. In the presented approach, the data set is augmentedwith directions in the tangent space of points in stress-strain space. Moreover, the data set isdivided into subsets corresponding to different material behavior, e.g. elastic and inelastic.Based on the classification, transition rules map the modeling points to the various subsets.The approach and its numerical performance will be demonstrated by applying it to modelsof non-linear elasticity and elasto-plasticity with isotropic hardening.
Keywords: data-driven computing, tangent space information, transition rules,inelasticity, data science
1. Introduction
The distance-minimizing data-driven computational method, introduced by Kirchdoer-fer and Ortiz [1], incorporates experimental material data into numerical calculations ofboundary-value problems, and therefore bypasses the empirical material modeling step. Inparticular, the optimization problem consists of calculating the closest point in the materialdata set consistent with the field equations of the problem, i.e., compatibility and equilibriumequations in continuum mechanics. ∗ Corresponding author
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[email protected] (K. Hackl ) a r X i v : . [ c s . C E ] J a n or a variety of elasticity problems the approach is elaborated and the associated conver-gence properties are well analyzed in [2, 3, 5, 6, 7]. However, problems arise when dealing withhistory-dependent data as present in inelastic materials, provides one uses nearest neighborclustering only. Therefore, local histories are included into the data set in [4]. Nonetheless, itis still necessary to resort to additional models for the evolution of internal variables. Thusa mixed formulation is obtained consisting of a combination of classical and data-drivenmodeling.This paper presents a new approach by augmenting the data sets with directions in thetangent space of points in stress-strain space. A similar second order data-driven scheme,formulated by [8], uses tensor voting [9] to obtain the point-wise tangent space. In contrast,the new approach includes the tangent space directly into the distance-minimization data-driven formulation, which leads to a much more concise system of equations. Furthermore,the integration of the tangent space enables interpolation in regions of sparse data sampling,whilst ensuring the internal states to cohere with the data set. An additional step to dealwith inelasticity is to classify the underlying data structure into subsets corresponding todifferent material behavior. Based on this, transition rules will be defined to map the internalstates of the system to the various subsets.To provide a general setting, Section 2 introduces the basic definitions and derivationof the classical distance-minimizing data-driven computing method. Section 3 presents theextension to inelasticity predicated on the extension of the data sets by tangent space in-formation and the classification of the data into subsets corresponding to different materialbehavior. Additionally tangential transition rules are defined to map the modeling points tothe various data subsets. Section 4 demonstrates the performance of the suggested methodvia numerical examples employing non-linear elasticity and elasto-plasticity with isotropichardening. At the end, Section 5 summarizes the results and gives recommendations con-cerning future research topics.
2. Data-driven computing paradigm
In the following the data-driven computing paradigm will be summarized for the readersconvenience based on the definitions and formulations in [1, 4]. Let Ω ⊂ R d with d ∈ N be a discretized system encountering displacements u = { u i ∈ R n i } ni =1 subjected to appliedforces f = { f i ∈ R n i } ni =1 , where n ∈ N is the number of nodes and n i the dimension at node i . The internal state is characterized by strain and stress pairs z e = ( ε e , σ e ) ∈ R d e with d e ∈ N being the dimension of stress and strain at material point e = 1 , . . . , m , where m ∈ N is the number of material points. The internal state of the system is subject to thecompatibility and equilibrium condition ε e = B e u e , ∀ e = 1 , . . . , m, (1) m (cid:88) e =1 w e B Te σ e = f . (2)2n this case w e is a positive weight and B e is a strain-displacement matrix. Defining z = { ( ε e , σ e ) } me =1 , the constraints (1) and (2) define a subspace C := (cid:110) z ∈ m ą e =1 R d e : (1) and (2) (cid:111) , (3)which is denoted as constraint set. Since the set is material-independent, the connectivitybetween ε e and σ e is still missing. Instead of using a functional relationship, the informationabout the material is given by means of a data set D := m ą e =1 D e with D e := { (ˆ ε i , ˆ σ i ) ∈ R d e } n e i =1 , (4)where n e ∈ N being the number of local data points; which classically consists of experimentalmeasurements or data achieved from small scale simulations. To define the data-drivenproblem the local space R d e will be metricized by means of norms (cid:107) z e (cid:107) e := (cid:18) (cid:107) ε e (cid:107) + 12 (cid:107) σ e (cid:107) (cid:19) / . (5)To avoid the dimensional dependency, the stresses are made dimensionless by the mapping( σ e , ˆ σ e ) → E e ( σ e , ˆ σ e ), with numerical scalar E e ∈ R + , typically being of the type of anelastic stiffness, e.g., a representative Young’s modulus. The corresponding local distancefunction d e ( z e , ˆ z e ) := (cid:107) z e − ˆ z e (cid:107) e (6)with z e , ˆ z e ∈ R d e , can be used to define a distance for z , ˆ z ∈ Ś me =1 R d e in the global spaceby d ( z , ˆ z ) := m (cid:88) e =1 d e ( z e , ˆ z e ) . (7)The distance-minimizing data driven problem, introduced by [1], readsmin ˆ z ∈D min z ∈C d ( z , ˆ z ) = min z ∈C min ˆ z ∈D d ( z , ˆ z ) , (8)i.e. the aim is to find the closest point consistent with the kinematics and equilibrium lawsto a material data set.The approach as well as the convergence and well-posedness have been studied on non-linear elastic material behavior (cf. [1, 3]). In the following the data-driven paradigm willbe extended to inelasticity. 3 . Extension to inelasticity In the following, we will suggest an extension of the data-driven paradigm to inelasticmaterials. This is a non-trivial task, since the same point in stress-strain space might corre-spond to different material behavior. Whereas it is proposed in [4] to include local historiesinto the data set, we will extend the data set by the tangent space information. Note thatmaterial tangents are accessible experimentally by appropriately varying the applied loading.For this purpose, let us introduce the extended data set D ∆ = m ą e =1 D ∆ e with D ∆ e := { (ˆ ε i , ˆ σ i ) + (∆ˆ ε i , ∆ ˆ σ i ) ∈ R d e } n e i =1 , (9)where ∆ˆ ε i , ∆ ˆ σ i ∈ R d e are tangential strain and stress increments satisfying the tangentialstiffness relation ∆ ˆ σ e = C e ∆ˆ ε e , (10)with symmetric and positive definite matrices C e ∈ R d e × d e . The actual independent data isthen given by (ˆ ε e , ˆ σ e , C e ), i.e. strain, stress and tangent space, whereas ∆ˆ ε e is a modelingvariable in the same sense as ε e and σ e , denoting the position on the tangent space at theindividual data point. The main idea is the evaluation of the data point ˆ z ∈ D closest tothe internal material point z ∈ C and additionally closest to the local tangential direction.Thus, the presented extension uses the underlying structure in order to remain as close aspossible to the local data sets. The definition of D ∆ allows to augment the local tangent spaces directly into the distance-minimization data-driven formulation. With optimal data points { ˆ z e ∈ D ∆ e } me =1 given, e.g.from a previous iteration, the minimization problem (8) can be written asMinimize m (cid:88) e =1 d e ( z e , ˆ z e )s.t. ε e = B e u e , m (cid:88) e =1 w e B Te σ e = f and ∆ ˆ σ e = C e ∆ˆ ε e . (11)The compatibility constraint can be enforced by expressing the material strains in termsof displacements i.e. z e = ( B e u e , σ e ). In addition the tangential stresses are expressed interms of tangential strains i.e. ˆ z e = (ˆ ε e , ˆ σ e ) + (∆ˆ ε e , C e ∆ˆ ε e ) ensuring the tangential stiffnessrelation. The equilibrium constraint can be enforced by means of Lagrange multipliers η ,leading to a Lagrangian of the form L = m (cid:88) e =1 d e ( z e , ˆ z e ) − (cid:32) m (cid:88) e =1 w e B Te σ e − f (cid:33) η . (12)4he necessary stationary conditions of L are given by δ u e : B Te ( B e u e − ˆ ε e − ∆ˆ ε e ) = , (13) δ σ e : ( σ e − ˆ σ e − C e ∆ˆ ε e ) = w e B e η e , (14) δ ∆ ˆ ε e : ( B e u e − ˆ ε e − ∆ˆ ε e ) + C e ( σ e − ˆ σ e − C e ∆ˆ ε e ) = , (15) δ η : m (cid:88) e =1 w e B Te σ e = f . (16)Multiplying equation (14) by matrix C e and using (15) gives( B e u e − ˆ ε e − ∆ˆ ε e ) + w e C e B e η e = . (17)In addition, multiplying (17) by B Te , using equation (13) and summing up over all materialpoints e = 1 , . . . , m leads to (cid:32) m (cid:88) e =1 w e B Te C e B e (cid:33) η = , (18)which implies η = . Consequently, equation (14) and (17) give an interpretation for thetangential increments ∆ˆ ε e = B e u e − ˆ ε e = ε e − ˆ ε e , (19)∆ ˆ σ e = C e ∆ˆ ε e = σ e − ˆ σ e . (20)Moreover, using relation (20) for equation (16) and reordering results in m (cid:88) e =1 w e B Te C e ∆ˆ ε e = f − m (cid:88) e =1 w e B Te ˆ σ e . (21)Finally substitution of (19) into (21) the corresponding Euler-Lagrange equations resultingto the following linear equation system: (cid:32) m (cid:88) e =1 w e B Te C e B e (cid:33) u = f − m (cid:88) e =1 w e B Te ( ˆ σ e − C e ˆ ε e ) . (22)The global state z ∈ C then follows by evaluating the local the strain increment ε e = B e u e , (23) σ e = ˆ σ e + C e ∆ˆ ε e = ˆ σ e + C e ( ε e − ˆ ε e ) , (24)for all material points e = 1 , . . . , m .It remains to determine the optimal local data points, i.e., the stress and strain pairs5 z e = (ˆ ε e , ˆ σ e ) in the local data sets D ∆ e that result in the closest possible satisfaction of com-patibility and equilibrium. The determination of the optimal points can be done iteratively.The iterations are performed until the distance d ( z , ˆ z ) is not lower than in the iteration be-fore or when a certain tolerance is reached. Due to the usage of the tangent-space structure,only only a few or even just one iteration are required. This constitutes a considerable in-crease of efficiency in comparison with the original algorithm as introduced in [1]. Moreover,because η = , only one global system of equations have to be solved instead of two in theoriginal algorithm. Figure 1: Visualization of data-driven method extended by tangent space. Modeling points z k +1 e minimizedistance to the tangent space associated with data points ˆ z ke , respecting compatibility and equilibriumconstraints. Data points ˆ z k +1 e minimize distance to modeling points z k +1 e . lgorithm 1 Data-driven solver for non-linear material behavior at loading step k + 1 Require: strain-displacement matrices { B e } me =1 , weights { w e } me =1 , load f k +1 , tolerance tol Data: data points { ˆ z ke } me =1 , tangent matrices { C ke } me =1 , data sets {D e } me =1 , function DDsolver ( {D e } me =1 , { C ke } me =1 , { ˆ z ke } me =1 , f k +1 ) while true do Solve equation system: (cid:32) m (cid:88) e =1 w e B Te C ke B e (cid:33) u k +1 = f k +1 − m (cid:88) e =1 w e B Te ( ˆ σ ke − C ke ˆ ε ke ) for e = 1 → m do ε k +1 e = B e u k +1 , σ k +1 e = ˆ σ ke + C ke ( ε k +1 e − ˆ ε ke ) end forfor e = 1 → m do min { d e ( z k +1 e , ˆ z e ) | ˆ z e ∈ D e } end forif d ( z k +1 , ˆ z ) ≤ tol thenbreakelse { ˆ z ke } me =1 ← { ˆ z e } me =1 end ifend whilereturn { z k +1 e } me =1 end function Given data points { ˆ z ke } me =1 and tangent matrices { C ke } me =1 at time t k and load f k +1 at time7 k +1 , the modeling points { z k +1 e } me =1 and data points { ˆ z k +1 e } me =1 are calculated. Fig. 1 givesa visualization of a single algorithmic loading step. The data-driven solver for non-linearmaterial behavior using fixed-point iteration is summarized in Algorithm 1. To simulate inelastic material behavior, the main task is to capture history dependence.This is achieved by associating different tangent spaces to data points with different history.Assuming an underlying data structure, as proposed in [8], the local material data sets D ∆ e are classified into subsets corresponding to different material behavior, e.g. elastic andinelastic: D ∆ e = ˙ (cid:91) p D ∆ , pe with p = { elastic , inelastic } . (25)Thus, data points with close or even the same strain and stress values may possess vastlydifferent tangent spaces; in the elastic case essentially determined by the elastic stiffnessand in the plastic case by the hardening modulus. It should be emphasized that it is easilypossible to distinguish experimentally between elastic and plastic material behavior. Basedon the classification, transition rules map the modeling points to the various subsets.In the following, a transition mapping is derived for the case of elasto-plasticity withisotropic hardening. The kinetics of elasto-plasticity is governed by a yield condition of theform σ com ( σ ) ≤ σ y , (26)where σ com ( σ ) is a comparison stress dependent on the current stress state, e.g. σ com ( σ ) = (cid:112) / (cid:107) dev σ (cid:107) in the case of von Mises ( J ) plasticity, and σ y denotes the yield stress, amaterial property depending on the loading history in the case of isotropic hardening. For σ com ( σ ) < σ y , we have elastic behavior, for σ com ( σ ) = σ y plastic behavior.Given values of modeling points { z k +1 e } me =1 using the data-driven algorithm 1, the tran-sition mapping for material state e = 1 , . . . , m at loading step k + 1 can be formulatedas:(i) assign local data set ˜ D e by˜ D e = D ∆ , pe with p ≡ (cid:40) elastic , if σ com ( σ k +1 e ) < σ y ,e inelastic , otherwise . (27)(ii) if p ≡ inelastic, set new yield stress at σ y ,e := σ com ( σ k +1 e );(iii) find closest data point ˆ z k +1 e in data set ˜ D e to modeling points z k +1 e withmin { d e ( z k +1 e , ˆ z k +1 e ) | ˆ z k +1 e ∈ ˜ D e } . (28)While step (i) maps the modeling points to the corresponding data sets, steps (ii) and(iii)define a new yield limit and find the closest data point inside these sets for the next loading8ncrement. These formulations give rise to corresponding representational scheme in thecontext of data-driven inelasticity, which are summarized in Algorithm 2. Algorithm 2
Data-driven transition rules for inelasticity at loading step k + 1 Require: load f k +1 , yield stresses { σ y ,e } me =1 Data: data points { ˆ z ke } me =1 , tangent matrices { C ke } me =1 , data sets { ˜ D e } me =1 , data subsets { ( D ∆ , elastic e , D ∆ , inelastic e ) } me =1 { z k +1 e } me =1 = DDsolver ( { ˜ D e } me =1 , { C ke } me =1 , { ˆ z ke } me =1 , f k +1 ) for e = 1 → m doif σ com ( σ k +1 e ) < σ y ,e then ˜ D e ≡ D ∆ , elastic e else ˜ D e ≡ D ∆ , inelastic e σ y ,e = σ com ( σ k +1 e ) end if min { d e ( z k +1 e , ˆ z k +1 e ) | ˆ z k +1 e ∈ ˜ D e } end for4. Numerical results for a 2D problem In this section the performance of the presented data-driven solver extended by thetangential space information will be illustrated in a typical benchmark example consideringa rectangular plate with a circular hole under loading (cf. Fig. 2). The plate has thedimensions of 1 m × . igure 2: Discretization and boundary conditions for a rectangular plate with a circular hole under loading. The following simulations deal with the analysis of a non-linear elastic material and anelasto-plastic von Mises material with isotropic hardening. Notice that the error between adata-driven solution z k and its corresponding reference solution z k, ref shall be calculated bymeans of the root-mean-square deviation of strain and stress defined byRMSD( z ) = (cid:80) Tk =0 Error( z k ) T , (29)where T ∈ N is the number of total loading steps, z ke = ( ε ke , σ ke ) the local data-driven statesand z k, ref e = ( ε k,refe , σ k, ref e ) the local reference states at step k ≤ T . The error is given byError( z k ) = (cid:80) me =1 w e (cid:107) z ke − z k, ref e (cid:107) (cid:80) me =1 w e (cid:107) z k, ref e (cid:107) , (30)with (cid:107) · (cid:107) given by definition (5). In the following, we are considering a non-linear elastic model. The material parametersof the reference solid used for the reference solution and data sets are Young’s modulus E = 100 · Pa, Poisson’s ratio ν = 0 . C = E (1 + ν )(1 − ν ) − ν ν ν − ν
00 0 (1 − ν ) . (31)The response is computed using a non-linear relation defined by σ = λf (tr( ε )) + µ ε + Cε with f ( x ) := c tanh( c x ), parameters c = 0 . c = 0 . · , identity matrix and Lam´econstants λ, µ defined by λ = Eν (1 + ν )(1 − ν ) and µ = E ν ) . (32)10 igure 3: Projection of the data set to the ε xx - σ xx -plane, according to non-linear elastic material behaviorwithin a normal and uniform distribution setting. For the data-driven computation two different types of data distributions are investigated.The first data set is created by a zero-mean normal distribution with a standard deviationof 0 .
001 and the second data set is created by a uniform distribution within [ − . , .
03] forstrains in each direction. Figure 3 visualizes both distribution settings. The apparent verticalalignment of the data set is an artifact produced by the projection onto the ε xx − σ xx − plane.Finally, the simulation of problem in Fig. 2 is performed by applying a load increasing from0 to 1 . · Pa with 100 incremental steps using a constant normalized time step of ∆ t = 1.Due to the random nature of the data distribution, each simulation returns a different error.To cover a wide spectrum of the errors produced, we run 100 simulations corresponding toindependent realizations of both distributions.11 a) (b)Figure 4: RMSD Error of data-driven solver for normal and uniform distributed data points. (a) Convergencewith respect to data size. (b) Dependency of the error on applied noising. The shaded areas show the spreadof the error arising from the different data set realizations. The error plot in Fig. 4a shows a linear rate of convergence, which corresponds to thedata-driven convergence analysis of elastic problems in [1]. Figure 4b shows the dependenceof the error from noising ranging from 1% to 10% of the maximum values of strains andstresses applied to the various data sets. The shaded areas show the spread of the errorarising from the different data set realizations used in the independent simulation runs.Note, that apparently data sets with normal distribution give better performance.
This example illustrates the performance of the data-driven method extended by tran-sition rules by considering an elasto-plastic von Mises material with isotropic hardening forthe boundary value problem in Fig. 2. The material parameters of the reference solid usedfor the reference solution and data sets are Young’s modulus E = 200 · Pa, Poisson’sratio ν = 0 .
3, isotropic hardening modulus H = E/
20, initial yield stress σ y0 = 250 · Paand elasticity matrix given by (31). The response is computed in a standard manner usinga J -plasticity model based on an iterative predictor-corrector return mapping algorithmembedded in a Newton-Raphson global loop to restore equilibrium.12 igure 5: A virtual test of a plate with random holes to generate suitable data sets.(a) Reference solution(b) Data-driven solutionFigure 6: Von Mises stress distribution at maximum loading at each Gaussian integration point using (a) J -plasticity model and (b) data-driven algorithm. Following [4], a virtual test employing the geometry depicted in Fig. 5 is used to generatean accurate coverage of suitable local material states and loading paths of various set sizes.For the data-driven simulation, the geometry shown in Fig. 2 is used again. The appliedload increases from 0 to 1 . · Pa, decreases to 0 and then increases again to 2 · Pa,using a constant normalized time step of ∆ t = 1.Figure 6 shows the data-driven solution at the maximum loading magnitude using adata sample containing 10 points. The convergence of the maximum displacement to thereference displacement based on a J (a)(b)Figure 7: Convergence property of the extended data-driven method using tangential transition rules forelasto-plastic material behaviour. (a) Maximum displacement (vertical displacement of lower right vertexversus traction (resultant load of right edge) for different data resolution. (b) RMSD Error for each dataresolution. The shaded area shows the deviation of the error arising from different independent virtual tests. . Conclusions We present an approach extending the model-free data-driven computing method of prob-lems in elasticity of Kirchdoerfer and Ortiz [1] to inelasticity. The original method usesnearest neighbor clustering and therefore challenges arise dealing with history-dependentdata. This issue is treated in this work by extending the formulation by including point-wisetangent spaces and classifying the data structure into subsets corresponding to different ma-terial behavior. Based on the classification, transition rules are defined to map the materialpoint to the classified data subsets, which incorporates with the idea that data points areconnected by an underlying structure to each other. Additionally, minimizing the distanceto local tangent spaces ensures data point connectivity and enables interpolation in regionslacking information of data.Furthermore, the presented scheme can be easily applied to non-linear elasticity as well,noticing that the resulting system of equations of the minimization problem is reduced,leading to greater efficiency. A numerical example has been presented to demonstrate theapplication to data-driven inelasticity and its numerical performance.Generally, it can be concluded that improvements in accuracy of the presented approachincrease for larger data sets and it correlates with the convergence analysis of data-drivenelasticity. Nevertheless, it should be mentioned that the ensurance of specific quality of thedata such as good coverage of material states and loading paths constitutes a critical issueconcerning the availability of real experimental data. Another issue concerns the classifi-cation of the data into subsets corresponding to material behavior. This could be done byefficient machine-learning algorithms e.g. spectral or density based clustering. These gen-eralizations of the data-driven paradigm suggest important directions for future research,especially the usage of machine-learning methods providing further improvement and au-tomation.
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