**1. Introduction**

Assessing the properties of mechanical structures with real physical experiments is expensive, as it costs both time and resources. To reduce these costs of knowledge enrichment in the field of structural analysis, computer simulations of structural mechanics have become crucial. An essential simulation method is the finite element method (FEM) in which the simulation domain space is represented by a finite number of connected elements. Space- and time-dependent behavior between connected elements and within the elements themselves is governed by physical equations. Observation of real physical experiments provides the coefficients for these governing equations. Since most geometries and use cases cannot be solved analytically, an approximation of the proposed physical equations is obtained by numerical methods [1]. However, solving complex problems with FEM is time-consuming and computationally expensive. In order to reduce the computational effort, surrogate modeling offers a promising solution [2].

Surrogate models are trained in a supervised manner and are designed to learn the function mapping between inputs and outputs from a given FEM simulation use case. With a sufficient amount of training data with respect to the use case, an according model is able to substitute for the FEM simulation use case up to a certain accuracy.

There is already a considerable number of related work concerning surrogate modeling of structural mechanics simulations with machine learning (ML) or deep learning (DL)

**Citation:** Hoffer, J.G.; Geiger, B.C.; Ofner, P.; Kern, R. Mesh-Free Surrogate Models for Structural Mechanic FEM Simulation: A Comparative Study of Approaches. *Appl. Sci.* **2021**, *11*, 9411. https://doi.org/10.3390/app11209411

Academic Editors: Jin-Gyun Kim, Jae-Hyuk Lim and Peter Persson

Received: 15 September 2021 Accepted: 3 October 2021 Published: 11 October 2021

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approaches. In the following, we want to present the most important works for this paper. Artificial neural networks (ANN) are used in the work of Roberts et al. [3] to predict damage development in forged brake discs reinforced with Al-SiC particles, using damage maps. The ANN is a multilayer perceptron (MLP), and training data are obtained from FEM simulations using the commercial DEFORM simulation software. For rapid estimation of forming and cutting forces for given process parameters, Hans Raj et al. [4] investigate a method using MLP models. The researchers focus on two processes: hot upsetting and extrusion. Each process, represented by a MLP, is trained with FEM simulation results from the FORGE2 commercial FEM simulation software. García-Crespo et al. [5] predict the projectile response after impact with steel armor using a MLP; their surrogate model studied is trained with data from FEM simulations of the use case. Nourbakhsh et al. [6] explore generalizable surrogate models for 3D trusses, using MLP and FEM training data. Chan et al. [7] estimate the performance of hot-forged product designs, using a MLP trained on FEM results obtained with the commercial software DEFORM. D'Addona and Antonelli [8] use single-layer feedforward ANNs instead of FEM as a metamodel in a sequential approximate optimization (SAO) algorithm. In a case study on hot forging of a steel disk, they compare their results with an ANN trained on FEM simulation results and the FEM simulation software QForm3D. Gudur and Dixit [9] predict the velocity field and location of neutral point of cold flat rolling with a MLP trained with rigid-plastic FEM simulation results. Pellicer-Valero et al. [10] predict the mechanical behavior of different livers with MLPs trained from FEM simulations.

Abueidda et al. [11] estimate the mechanical properties of a two-dimensional checkerboard composite using a convolutional neural network (CNN) trained with FEM results. Regarding mesh-based approaches, Pfaff et al. [12] present a framework to train graph neural networks (GNN) on mesh-based simulations and show the applicability in aerodynamics, structural mechanics, and fabric.

Surrogate models were also obtained using classical, i.e., non-neural ML, approaches. For example, the authors of [3] apply Gaussian process regression (GPR) besides ANN in their approach. Loghin and Ismonov [13] predict the stress intensity factors, using GPR trained with FEM results of a classical bolt-nut assembly. Ming et al. [14] model the electrical discharge machining process with GPR trained from data generated with numerical FEM simulation.

Using support vector regression (SVR), Pan et al. [15] construct a metamodel in an optimization approach for lightweight vehicle design. Training data are generated, using design of experiment approaches with FEM simulations. To predict the stress at the implant– bone interface, Li et al. [16] utilize SVR in order to replace FEM simulation. Hu and Li [17] estimate cutting coefficients in a mechanistic milling force model with SVR trained with FEM simulation data.

Employing tree-based models, Martínez-Martínez et al. [18] estimate the biomechanical behavior of breast tissue under compression, using three different tree-based models trained from FEM simulations. The models are trained with FEM data in terms of nodal coordinates and nodal tissue membership. Zhang et al. [19] estimate the base failure stability for braced excavations in anisotropic clay using extreme gradient boosting, random forest regression (RFR) and data obtained from FEM simulation results. Qi et al. [20] utilize a decision tree regressor to predict the mechanical properties of carbon fiber reinforced plastics with data obtained from FEM simulations. Besides MLPs Pellicer-Valero et al. [10] utilize RFRs to predict the biomechanics of livers.

A recent neural network–based approach are physics informed neural networks (PINNs). PINNs are trained simultaneously on data and governing differential equations and can be used for the solution and inversion of equations governing physical systems. Utilizing PINNs, Haghighat and Juanes [21] substitute a particular FEM simulation of a perforated strip under uniaxial extension. In [22], Haghighat et al. present a surrogate modeling approach with PINNs and a specific use case. Focusing on consistency, Shin [23] evaluates findings regarding PINNs with Poisson's equation and the heat

equation. Yin et al. [24] use PINNs to predict permeability and viscoelastic modulus from thrombus deformation data, described by the fourth-order Cahn–Hilliard and Navier–Stokes equations. In addition to the application of PINNs in structural mechanics problems, there is also a considerable number of papers, especially in computational fluid dynamics [25–29].

Related work shows capabilities of surrogate modeling, thus demonstrating the feasibility of supervised learning models trained with FEM simulations. From our analysis of the existing literature, we identify the following drawbacks:


1. We present the main DL and ML methods together with a compact description


With our work, we pave the way of mesh-free surrogate modeling for practical use: we provide a basis for efficient model and hyperparameters selection regarding use case and performance metrics. These insights shall not only assist the domain expert during model selection, but will also help in consolidating the current research in mesh-free surrogate modeling for structural mechanics applications.

We report all information to make our experiments reproducible. If certain model settings are not mentioned, they are left at default values. Moreover, our FEM simulations are performed with Abaqus Student Edition 2019 (Dassault Systèmes, Velizy-Villacoublay, France), and thus, the process of data generation is not limited to commercial software, which makes it possible for everyone to connect to our research.

The remainder of this paper is organized as follows. In Section 2, we present the materials and methods of our experiments, first providing insights into the process of data generation, using the FEM simulations in Section 2.1, then describing the datasets obtained from the FEM simulations in Section 2.2, followed by the ML and DL models used in Section 2.3. Section 3 shows the results, which are discussed in Section 4. In Section 5, we present the conclusion of our work and an outlook for the future.

#### **2. Materials and Methods**

In this section, we present all relevant information about the methodology of our experiments. First, Section 2.1 provides an overview of the data generation process, using three classic FEM simulation use cases. Then, Section 2.2 describes the datasets used from the FEM simulations, and Section 2.3 presents the ML and DL models used. A more detailed overview of the mathematical background and assumptions of the ML and DL models can be found in the Appendix. When predicting a particular use case with a surrogate model, the individual nodes discretizing the particular geometry of the use case (i.e., mesh) are sequentially input into the surrogate model with the appropriate generalization variable. The surrogate model then predicts the output of each node in sequence; see Figure 1.

**Figure 1.** Principle of our surrogate model approach: all *N* nodes (i.e., their coordinates), together with the respective generalization variable, are sequentially entered into a surrogate model, which then sequentially predicts the outcome of the respective coordinates (i.e., the displacements, strains, and stresses of the respective node).

It should be noted that there are no constraints on the discretization (mesh), i.e., the node coordinates can be freely chosen within the simulation domain and nodes are not connected to each other. Therefore, we refer to our approach as mesh-free, but we want to clearly distinguish ourselves from other mesh-free methods, such as smoothed particle hydrodynamics, the diffuse element method, the moving particle finite element method, etc. The predictions of the individual nodes together constitute the prediction for the simulation domain of the particular use case. By adding the nodal displacement outputs of the surrogate model to the initial node coordinates, we obtain the new deformed geometry. Further surrogate model outputs (e.g., stresses, strains) describe the queried nodes and thus the complete simulation domain in more detail.

#### *2.1. FEM Use Cases*

For illustration, we base our evaluation on three classic use cases from structural mechanics. We consider the (1) tensile load, (2) bending load and (3) compressive load:


See Table 1 and Figure 2. We utilize an isotropic elasto-plastic rate-independent material model (i.e., a perfectly plastic material). The kinematic relations for our 2D plane strain use cases are defined by the total strain components *εxx* = *<sup>∂</sup>ux <sup>∂</sup><sup>x</sup>* , *<sup>ε</sup>yy* <sup>=</sup> *<sup>∂</sup>uy <sup>∂</sup><sup>y</sup>* , *εxy* = <sup>1</sup> <sup>2</sup> ( *<sup>∂</sup>ux <sup>∂</sup><sup>y</sup>* <sup>+</sup> *<sup>∂</sup>uy <sup>∂</sup><sup>x</sup>* ), *εzz* = 0 with displacements *ux* and *uy* and deviatoric strain components *exx* <sup>=</sup> *<sup>ε</sup>xx* <sup>−</sup> *<sup>ε</sup>vol* <sup>3</sup> , *eyy* <sup>=</sup> *<sup>ε</sup>yy* <sup>−</sup> *<sup>ε</sup>vol* <sup>3</sup> , *exy* <sup>=</sup> *<sup>ε</sup>xy* and *ezz* <sup>=</sup> <sup>−</sup>*εvol* <sup>3</sup> . Since there is no volumetric plastic strain in the von Mises yield function, the volumetric strain can be expressed as

*εvol* = trace(*ε*) s.t. *εvol* = *εxx* + *εyy*. The deviatoric stress components are defined by *sxx* = *σxx* − ( *σxx*+*σyy*+*σzz* <sup>3</sup> ), *syy* = *σyy* − ( *σxx*+*σyy*+*σzz* <sup>3</sup> ), *sxy* = *σxy* and *szz* = *σzz* − ( *σxx*+*σyy*+*σzz* <sup>3</sup> ), where *σij* (*i*, *j* ∈ {*x*, *y*}) are the components of the Cauchy stress tensor. The plastic strain components are defined by *ε pl xx* = *ε*¯*pl* <sup>3</sup> 2 *sxx <sup>q</sup>* , *ε pl yy* = *ε*¯*pl* <sup>3</sup> 2 *syy <sup>q</sup>* , *ε pl xy* = *ε*¯*pl* <sup>3</sup> 2 *sxy <sup>q</sup>* and *ε pl zz* = *ε*¯*pl* <sup>3</sup> 2 *szz q* with equivalent plastic strain of the von Mises model as *<sup>ε</sup>*¯*pl* <sup>=</sup> *<sup>ε</sup>*¯ <sup>−</sup> *<sup>σ</sup><sup>Y</sup>* <sup>3</sup>*<sup>μ</sup>* ≥ 0, where *σ<sup>Y</sup>* is the yield stress and *μ* the second Lamé parameter. The total equivalent strain is defined by *ε*¯ = 2 <sup>3</sup> ∑*i*,*j*∈{*x*,*y*} *eijeij* with deviatoric strain components *eij*. The decomposition of the strain is *εij* = *εel ij* + *ε pl ij* with elastic component *<sup>ε</sup>el ij* and plastic component *ε pl ij* of the respective strain matrices. The equivalent stress is defined by *q* = 3 <sup>2</sup> *sijsij*. In our PINN approach, we utilize the definitions of the total strain components, deviatoric strain and stress components and plastic strain components in the respective regularization term.

We use quarter symmetry in use cases 1 and 3 to make efficient use of computational resources. Additional information regarding the variation of parameters in the simulations is presented in Table 2, where simulations marked in bold are used for the test and evaluation of the surrogate models and are not in the training dataset. Conversely, simulations not marked in bold represent the training dataset and are not in the test dataset. In use cases exhibiting varied geometry parameters (i.e., elongation of a plate and compression of a block use cases), the mesh is also different in each simulation. Thus, we train and evaluate the surrogate models on use cases with different meshes (i.e., in each simulation, the node coordinates differ).

**Table 1.** Classic FEM use cases. Overview of the three use cases and their main change and types of deformations. In the first two use cases, only a single change is conduced, while in the last use case, a combination of changes is studied.


The first use case, a perforated steel strip under tensile load, is similar to the nonlinear solid mechanics use case of [21,22]. However, in our approach, we evaluate the generalization over the perforation diameter and use material properties for steel and a top edge displacement of 5 mm in positive *y*-axis to consider a more challenging use case.

We execute different simulation settings, where the generalization variable (diameter of perforation) is changed in each simulation; see Figure 2a and Table 2. In our second use case, we simulate a bending beam that end is displaced about 5 mm in the positive xdirection; see Figure 2b. We vary the yield stress generalization variable in each simulation setting; see Table 2. In our third use case, we simulate a quarter-symmetric block with four perforations under compressive load, which is compressed about 5 mm in the negative y-axis; see Figure 2c. In this use case, we vary two generalization variables (yield stress and width of the block) in each simulation; see Table 2.

We evaluate our models on interpolation (i.e., that the generalization variables for testing are within the range of the generalization variables observed during training) and extrapolation (i.e., that the generalization variables for testing are outside the range of the generalization variables observed during training) tasks. In Table 2, we mark interpolation tasks with superscript (*i*) and extrapolation tasks with superscript (*e*).

In Figure 3, we present the perfect nonlinear elastoplastic material behavior of our use cases. The Young's modulus is 210 GPa, Poisson's ratio 0.3 and the yield stress 900 MPa. In our first use case, the perforated plate, we use this setting in each simulation. In the other two use cases, the yield stress varies, while the remaining material parameters stay the same.

**Figure 2.** The three use cases: (**a**) elongation of a plate (diameter = 100 mm) about 5 mm at the top end in positive y-direction, (**b**) bending of a beam by a displacement at the top end about 5 mm in positive x-direction, (**c**) compression of a block with four perforations in the center of the quarter-symmetric parts (width = 220 mm) about 5 mm in negative y-direction and (**d**) the considered coordinate system.

All parts are meshed, using plane strain 4-node bilinear quadrilateral elements with reduced integration and hourglass control. Please note that although [22] recommends the use of larger order elements for the approximation of body forces, we use bilinear elements since we do not use body forces in our surrogate modeling approaches. We create a finer mesh near additional geometric details (i.e., perforations in the plate and block use cases) and seed the perforation edge of the plate with an approximate size of 3.8 mm and the remaining edges with an approximate size of 5 mm. The perforation edges of the block are seeded with an approximate size of 3 mm and the remaining edges with an approximate size of 4 mm. The beam exhibits no comparable geometric details; thus, we seed all edges with an approximate size of 1.5 mm.

**Table 2.** Dataset generation by executing several different simulations with varying generalization variables (Plate: perforation *Diameter*, Beam: *Yield Stress* and Block: *Yield Stress* and *Width*), bold marked simulations are not in the training dataset and only used for test and evaluation. Interpolation tasks are marked with superscript (*i*) and extrapolation tasks with superscript (*e*).


**Figure 3.** Perfect nonlinear elastoplastic material properties for a Young's modulus of 210 GPa, Poisson's ratio of 0.3 and yield stress of 900 MPa. The yield stress varies in simulations regarding the beam and block use cases.

We obtain our FEM simulation results in the context of general static simulations. Details of the simulation steps are shown in Table 3. Simulation control parameters that are not listed are left at default values.

**Table 3.** Abaqus FEM simulation control parameters.

