PAMM
Volume 23, Issue 4 e202300079
RESEARCH ARTICLE
Open Access

Reduced order modeling of structural problems with damage and plasticity

Jannick Kehls, 

Corresponding Author

Jannick Kehls

Institute of Applied Mechanics, RWTH Aachen University, Aachen, Germany

Correspondence

Jannick Kehls, Institute of Applied Mechanics, RWTH Aachen University, Mies-van-der-Rohe-Str. 1, 52074 Aachen, Germany.

Email: [email protected]

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Steffen Kastian, 

Steffen Kastian

Institute of Applied Mechanics, RWTH Aachen University, Aachen, Germany

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Tim Brepols, 

Tim Brepols

Institute of Applied Mechanics, RWTH Aachen University, Aachen, Germany

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Stefanie Reese, 

Stefanie Reese

Institute of Applied Mechanics, RWTH Aachen University, Aachen, Germany

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First published: 20 November 2023
https://doi.org/10.1002/pamm.202300079
Citations: 1

Abstract

Model order reduction (MOR) techniques are used across the engineering sciences to reduce the computational complexity of high-fidelity simulations. MOR methods reduce the computation time by representing the problem using a lower number of degrees of freedom (DOF). The use of reduced order models (ROM) in the analysis of structural problems with damage and plasticity has the potential to significantly reduce computational time and increase efficiency. Of course, the approximation of a problem in a lower dimensional space introduces an approximation error that needs to be kept small enough so that the results of the ROM maintain their validity. One well-known reduced order modeling approach is the proper orthogonal decomposition (POD). POD is used to extract the dominant modes of the structure, which are then used to solve a problem in the smaller dimensional subspace. To overcome the limitations of the POD regarding nonlinear problems, the discrete empirical interpolation method (DEIM) is employed. An exemplary uncertainty quantification application is used to investigate the methodology. The investigation shows that the POD-based DEIM can significantly reduce the computational effort of highly nonlinear structural simulations incorporating damage while maintaining a high approximation accuracy.

1 INTRODUCTION

The increase of complex large-scale finite element simulations in recent years has simultaneously increased the demand for sophisticated model order reduction (MOR) techniques. Looking at the field of structural mechanics, one of the major MOR techniques is the proper orthogonal decomposition (POD), which is snapshot-based, meaning that a priori knowledge of the system response is used to create a smaller dimensional subspace in a so-called offline phase. In the online phase of the simulation, the problem is projected onto the subspace and can be solved there much faster than in the original space. Solving linear problems, the POD can drastically reduce the computational complexity of a model. Working with nonlinear problems, the POD has some major drawbacks. As the problem needs to be linearized and solved for example with the Newton-Raphson method, the system of equations has to be set up in every iteration step. With the complex geometries and materials used in modern structural simulations, the construction of the tangent matrix and the residual vector takes a large part of the simulation time. Therefore, different so-called hyper-reduction methods are being actively developed, investigated, and refined. Currently, promising methods are for example the empirical cubature (EC) (e.g. [1]), the Gauss-Newton with approximated tensors (GNAT) method (e.g. [2]), and the discrete empirical interpolation method (DEIM) (e.g. [3]). All of these hyper-reduction methods are based on the POD and generally aim to approximate the constantly recomputed quantities before projecting them onto the subspace in which the system of linear equations is solved. In this contribution, the DEIM is studied for highly nonlinear structural problems with damaging material behavior. To conduct reduced order simulations in the context of damaging material behavior, the reduced arc-length method is employed and investigated. An exemplary application in the field of uncertainty quantification is used to examine the accuracy, speedup potential, and stability of the reduced order models (ROM).

2 PROBLEM FORMULATION

The general displacement-based finite element approach for quasi-static nonlinear problems leads to the equation
G ( U ) : = R ( U ) − P = 0 , $$\begin{equation} \bm {G}(\bm {U}) := \bm {R}(\bm {U}) - \bm {P}= \mbox{$\bf 0$}, \end{equation}$$ (1)
with G ( U ) $\bm {G}(\bm {U})$ being the residual vector, R ( U ) $\bm {R}(\bm {U})$ being the internal force vector and P $\bm {P}$ being the external force vector. Using n to describe the total number of DOFs of the problem, all of these vectors have the dimension n × 1 $n \times 1$ . The internal force vector is dependent on the solution state U $\bm {U}$ , which makes the problem generally nonlinear. As Equation (1) can therefore not be solved directly, the Newton-Raphson scheme is employed leading to the iterative solution scheme
K T ( U ( i ) ) Δ U ( i + 1 ) = − G ( U ( i ) ) U ( i + 1 ) = U ( i ) + Δ U ( i + 1 ) G ( U ( i ) ) T Δ U ( i + 1 ) ≤ t o l e r a n c e i = i + 1 $$\begin{equation} \begin{aligned}\mathbf {K}_{T}(\bm {U}{}^{(i)}) \, \Delta \bm {U}{}^{(i+1)}&= - \bm {G}(\bm {U}{}^{(i)})\\ \bm {U}{}^{(i+1)}&= \bm {U}{}^{(i)}+ \Delta \bm {U}{}^{(i+1)}\\ \bm {G}(\bm {U}{}^{(i)}) ^T\: \Delta \bm {U}{}^{(i+1)}&\le tolerance\\ i &= i + 1 \end{aligned} \end{equation}$$ (2)
K T $\mathbf {K}_{T}$ describes the tangential stiffness matrix of the system, and Δ U ( i + 1 ) $\Delta \bm {U}{}^{(i+1)}$ is the incremental solution vector, for which the equation is solved for until the desired convergence criterion is met. Using the micromorphic approach described in ref. [4] to introduce a nonlocal damage degrees of freedom (DOF) which is coupled with its local counterpart by a penalty parameter, the quantities in the above solution scheme are further specified as:
K u u K u D ¯ K D ¯ u K D ¯ D ¯ ︸ K T Δ u Δ D ¯ ︸ Δ U = − G u G D ¯ ︸ G . $$\begin{equation} \begin{aligned}\underbrace{{\left[ \def\eqcellsep{&}\begin{array}{l l}\mathbf {K}_{uu} & \mathbf {K}_{u \bar{D}} \\[3pt] \mathbf {K}_{\bar{D}u} & \mathbf {K}_{\bar{D}\bar{D}} \end{array} \right]}}_{\mathbf {K}_T} \underbrace{{\left\lbrace \def\eqcellsep{&}\begin{array}{l}\Delta \bm {u} \\[3pt] \Delta \bar{\bm {D}} \end{array} \right\rbrace} }_{\Delta \bm {U}} = - \underbrace{{\left\lbrace \def\eqcellsep{&}\begin{array}{l}\bm {G}_u \\[3pt] \bm {G}_{\bar{D}} \end{array} \right\rbrace} }_{\bm {G}}. \end{aligned} \end{equation}$$ (3)
K u u $\mathbf {K}_{uu}$ describes the stiffness obtained from the mechanical problem, whereas K D ¯ D ¯ $\mathbf {K}_{\bar{D}\bar{D}}$ describes the stiffness coming from the micromorphic approach. K u D ¯ $\mathbf {K}_{u \bar{D}}$ and K D ¯ u $\mathbf {K}_{\bar{D}u}$ are mixed terms that are not necessarily equal to each other, generally leading to an asymmetric tangent stiffness matrix. Δ u $\Delta \bm {u}$ and Δ D ¯ $ \Delta \bar{\bm {D}}$ are the unknown displacements and the unknown nonlocal damage, respectively. Lastly, the residual G $\bm {G}$ consists of the residual of the displacement DOFs G u $\bm {G}_u$ as well as the residual of the nonlocal damage DOFs G D ¯ $\bm {G}_{\bar{D}}$ . For the detailed description and derivation of the model, the reader is referred to ref. [5].

3 POD FOR STRUCTURAL MECHANICS

For later purposes, Equation (1) is split into its linear and nonlinear parts according to ref. [6], such that it reads
G ( U ) : = R ( U ) − P = K l i n U + R n l ( U ) − P = 0 . $$\begin{equation} \begin{aligned}\bm {G}(\bm {U}) := & \bm {R}(\bm {U})-\bm {P}\\ =&\mathbf {K}_{lin} \, \bm {U}+\bm {R}_{nl} (\bm {U})-\bm {P}=\mbox{$\bf 0$}. \end{aligned} \end{equation}$$ (4)
K l i n $\mathbf {K}_{lin}$ now represents only the linear part of the stiffness matrix which is of course constant throughout the simulation. Multiplying with the solution vector, the linear part of the internal force vector is obtained. The nonlinear part is depicted by R n l ( U ) $\bm {R}_{nl}(\bm {U})$ and has to be recomputed in every iteration step due to the still-existing dependence on the solution state. This leads to the iterative solution scheme as follows:
( K l i n + K T n l ( U ( i ) ) ) Δ U ( i + 1 ) = − G ( U ( i ) ) U ( i + 1 ) = U ( i ) + Δ U ( i + 1 ) G ( U ( i ) ) T Δ U ( i + 1 ) ≤ t o l e r a n c e i = i + 1 . $$\begin{equation} \begin{aligned}(\mathbf {K}_{lin} + \mathbf {K}_{T \: nl}(\bm {U}{}^{(i)})) \, \Delta \bm {U}{}^{(i+1)}&= - \bm {G}(\bm {U}{}^{(i)})\\ \bm {U}{}^{(i+1)}&= \bm {U}{}^{(i)}+ \Delta \bm {U}{}^{(i+1)}\\ \bm {G}(\bm {U}{}^{(i)}) ^T\: \Delta \bm {U}{}^{(i+1)}&\le tolerance\\ i &= i + 1. \end{aligned} \end{equation}$$ (5)
The newly introduced quantity K T n l ( U ( i ) ) $\mathbf {K}_{T \: nl}(\bm {U}{}^{(i)})$ consists of only the nonlinear part of the stiffness matrix, but as it cannot be computed directly in real simulations, it is obtained by constructing the full tangential stiffness matrix first and subtracting the constant linear part of the stiffness matrix from it. Up to this point, the split into a linear and a nonlinear part does not provide any benefit, but as this split is necessary for the developments later, it is already introduced here for consistency and comprehensibility.
Assuming a n × m $n \times m$ dimensional projection matrix Φ m $\mathbf {\Phi }_m$ can be found, a Galerkin-projection leads to the approximation of the solution vector U $\bm {U}$ by
U ≈ U ¯ = Φ m U r e d , $$\begin{equation} \bm {U}\approx \bar{\bm {U}}= \mathbf {\Phi }_m \, \bm {U}_{red}, \end{equation}$$ (6)
where U r e d $\bm {U}_{red}$ is the m × 1 $m \times 1$ reduced solution vector. Then, Equation (4) can be rewritten as
G r e d ( U ¯ ) : = K l i n r e d U r e d + R n l r e d ( U ¯ ) − P r e d = 0 , $$\begin{equation} \begin{aligned}\bm {G}_{red}(\bar{\bm {U}}) := \mathbf {K}_{lin \: red} \: \bm {U}_{red}+ \bm {R}_{nl \: red}(\bar{\bm {U}}) - \bm {P}_{red}= \mbox{$\bf 0$}, \end{aligned} \end{equation}$$ (7)
with K l i n r e d : = Φ m T K l i n Φ m $\mathbf {K}_{lin \: red} := \mathbf {\Phi }_m^{T}\: \mathbf {K}_{lin}\: \mathbf {\Phi }_m$ , R n l r e d : = Φ m T R n l ( U ¯ ) $\bm {R}_{nl \: red} := \mathbf {\Phi }_m^{T}\: \bm {R}_{nl}(\bar{\bm {U}})$ and P r e d : = Φ m T P $\bm {P}_{red}:= \mathbf {\Phi }_m^{T}\: \bm {P}$ . The linearization leads to an order-reduced solution scheme that looks as follows:
( K l i n r e d + Φ m T K T n l ( U ¯ ( i ) ) Φ m ) Δ U r e d ( i + 1 ) = − G r e d ( U ¯ ( i ) ) U ¯ ( i + 1 ) = U ¯ ( i ) + Φ m Δ U r e d ( i + 1 ) G r e d T ( U ¯ ( i ) ) Δ U r e d ( i + 1 ) ≤ t o l e r a n c e i = i + 1 . $$\begin{equation} \begin{aligned}(\mathbf {K}_{lin \: red}+ \mathbf {\Phi }_m^{T}\: \mathbf {K}_{T \: nl}(\bar{\bm {U}}{}^{(i)}) \: \mathbf {\Phi }_m) \: \Delta \bm {U}_{red}^{(i+1)} &= - \bm {G}_{red}(\bar{\bm {U}}{}^{(i)})\\ \bar{\bm {U}}{}^{(i+1)}&= \bar{\bm {U}}{}^{(i)}+ \mathbf {\Phi }_m\Delta \bm {U}_{red}^{(i+1)}\\ \bm {G}_{red}^T(\bar{\bm {U}}{}^{(i)}) \: \Delta \bm {U}_{red}^{(i+1)} &\le tolerance\\ i &= i + 1. \end{aligned} \end{equation}$$ (8)
The stiffness matrix is in this case m × m $m \times m$ dimensional and the system of equations is solved in the reduced order space spanned by the projection matrix Φ m $\mathbf {\Phi }_m$ . This significantly reduces the computational complexity of solving the system. After the solution is found, the incremental solution vector is simply projected back to the full space and can be added to the total solution vector.
Using the POD, the projection matrix is obtained by carrying out a singular value decomposition (SVD) of the so-called snapshot matrix D U $\bm {D}_{U}$ . The snapshot matrix is created by gathering l sample solution vectors of the full problem such that D U = [ U 1 , U 2 , … , U l ] $\bm {D}_{U}=[\bm {U}_1, \bm {U}_2, \ldots, \bm {U}_l]$ . The sample solutions can for example be the solutions to the problem at various (pseudo-)timesteps, with different material parameters or even varying geometries. In principle, the sample vectors are only restricted by the vectors' dimension which have to be of the same size n. Having created the snapshot matrix, it is decomposed such that D U = Φ Σ V $\bm {D}_{U}=\mathbf {\Phi }\mathbf {\Sigma }\bm {V}$ . The matrix containing the left singular vectors is defined by Φ = [ ϕ 1 , ϕ 2 , … , ϕ n ] $\mathbf {\Phi }=[\bm {\phi }_1, \bm {\phi }_2, \ldots, \bm {\phi }_n]$ , whereas the matrix of right singular vectors is defined as V = [ v 1 , v 2 , … , v l ] ${\bm {V}}=[\bm {v}_1, \bm {v}_2, \ldots, \bm {v}_l]$ . The matrix of the singular values consists of only zero entries, except for the diagonal elements Σ i i $\Sigma _{ii}$ which are arranged in a decreasing order such that
Σ = Σ 11 ⋱ Σ n n 0 with Σ 11 ≥ ⋯ ≥ Σ n n . $$\begin{align} \mathbf {\Sigma }= \def\eqcellsep{&}\begin{pmatrix} \Sigma _{11} & &{} \\ & \ddots & {}\\ & & \Sigma _{nn} \\ & \mbox{$\bf 0$}&{} \end{pmatrix} \, \, \text{with} \, \, \Sigma _{11} \ge \dots \ge \Sigma _{nn} \, . \end{align}$$ (9)
The singular values are of great interest to us, as they can be used to estimate the amount of left singular vectors that are needed to capture the characteristics of the problem. The projection matrix containing the m most dominant left singular vectors (in accordance with the literature these are from here on called “POD modes”) is therefore obtained by truncating Φ $\mathbf {\Phi }$ after POD mode m such that we finally arrive at the n × m $n \times m$ dimensional projection matrix Φ m $\mathbf {\Phi }_m$ .

4 DEIM FOR STRUCTURAL MECHANICS

One major problem of the POD in the field of structural mechanics is the fact that the internal force vector is dependent on the solution state, making the problem nonlinear. As described before, the solution is therefore computed in an iterative manner which leads to the fact that the unreduced residual, as well as the unreduced stiffness matrix, have to be recomputed in every iteration step. As the material model becomes more complex, the computation of the stiffness matrix can become the most time-consuming part of the simulation. A solution to this problem are so-called hyper-reduction techniques. The general idea of these methods is that the stiffness matrix is approximated or interpolated by evaluating only selected elements or interpolation points. Going into detail about all the different hyper-reduction techniques is beyond the scope of this work, which is why only the mathematical basis of the DEIM is presented. To start off, a second projection matrix Ω k $\mathbf {\Omega }_k$ with dimensions n × k $n \times k$ is introduced which lets us approximate the nonlinear part of the internal force vector as
R n l ( U ) ≈ R ¯ n l ( U ) = Ω k R n l r e d ( U ) . $$\begin{equation} \bm {R}_{nl}(\bm {U}) \approx \bar{\bm {R}}_{nl}(\bm {U}) = \mathbf {\Omega }_k\: \bm {R}_{nl \: red}(\bm {U}). \end{equation}$$ (10)
The DEIM projection matrix Ω k $\mathbf {\Omega }_k$ is computed analogously to the POD projection matrix Φ m $\mathbf {\Phi }_m$ by carrying out a SVD and truncating the left singular vectors. To get an appropriate projection matrix, the snapshot matrix is this time constructed by gathering the nonlinear internal force vectors from pre-computed simulations so that it can be described by D R = [ R n l ( U 1 ) , R n l ( U 2 ) , ⋯ , R n l ( U l ) ] $\mathbf {D}_R=[\bm {R}_{nl}(\bm {U}_1) , \bm {R}_{nl}(\bm {U}_2) , \dots , \bm {R}_{nl}(\bm {U}_l)]$ .
The DEIM projection matrix is further utilized to obtain in a sensible manner interpolation DOFs that play an important role in the nonlinearity of the problem. A greedy algorithm used for this purpose was developed by Chaturantabut and Sorensen [3] and is a significant improvement over the gappy POD approach [7], where the selection of the interpolation DOFs is not as sophisticated and results generally in larger approximation errors. The interpolation matrix Z $\mathbf {Z}$ contains k columns with n entries. Each column contains a single non-zero entry that is equal to 1, corresponding to a DOF of the structure. Having defined the interpolation matrix, a multiplication of its transpose from the left with Equation (10) leads to the new expression for R n l r e d $\bm {R}_{nl \: red}$ :
Z T R n l ( U ¯ ) = Z T Ω k R n l r e d ( U ¯ ) R n l r e d ( U ¯ ) = ( Z T Ω k ) − 1 Z T R n l ( U ¯ ) $$\begin{equation} \begin{aligned}\mathbf {Z}^T\bm {R}_{nl}(\bar{\bm {U}}) &= \mathbf {Z}^T\: \mathbf {\Omega }_k\: \bm {R}_{nl \: red}(\bar{\bm {U}}) \\ \bm {R}_{nl \: red}(\bar{\bm {U}}) &= (\mathbf {Z}^T\: \mathbf {\Omega }_k)^{-1} \: \mathbf {Z}^T\bm {R}_{nl}(\bar{\bm {U}}) \end{aligned} \end{equation}$$ (11)
The k × k $k \times k$ dimensional matrix Z T Ω k $\mathbf {Z}^T\: \mathbf {\Omega }_k$ is ensured to be nonsingular. Considering Equations (10) and (11)2, the approximation of the nonlinear term can therefore be rewritten as
R n l ( U ¯ ) ≈ R ¯ n l ( U ¯ ) = Ω k ( Z T Ω k ) − 1 Z T R n l ( U ¯ ) . $$\begin{equation} \bm {R}_{nl}(\bar{\bm {U}}) \approx \bar{\bm {R}}_{nl}(\bar{\bm {U}}) = \mathbf {\Omega }_k(\mathbf {Z}^T\: \mathbf {\Omega }_k)^{-1} \: \mathbf {Z}^T\bm {R}_{nl}(\bar{\bm {U}}). \end{equation}$$ (12)
Inserting the above statement into the already POD-reduced Equation (7) leads to the hyper-reduced form of the residual
G r e d ( U ¯ ) : = K l i n r e d U r e d + Φ m T R ¯ n l ( U ¯ ) − Φ m T P = K l i n r e d U r e d + Φ m T Ω k ( Z T Ω k ) − 1 Z T R n l ( U ¯ ) − Φ m T P = K l i n r e d U r e d + M D E I M Z T R n l ( U ¯ ) − Φ m T P = 0 . $$\begin{equation} \begin{aligned}\bm {G}_{red}(\bar{\bm {U}}) := &\mathbf {K}_{lin \: red}\: \bm {U}_{red}+ \mathbf {\Phi }_m^{T}\: \bar{\bm {R}}_{nl}(\bar{\bm {U}})-\mathbf {\Phi }_m^{T}\: \bm {P}\\ =& \mathbf {K}_{lin \: red}\: \bm {U}_{red}+ \mathbf {\Phi }_m^{T}\: \mathbf {\Omega }_k(\mathbf {Z}^T\: \mathbf {\Omega }_k)^{-1} \: \mathbf {Z}^T\bm {R}_{nl}(\bar{\bm {U}}) - \mathbf {\Phi }_m^{T}\bm {P}\\ =& \mathbf {K}_{lin \: red}\: \bm {U}_{red}+ \mathbf {M}_{DEIM}\: \mathbf {Z}^T\bm {R}_{nl}(\bar{\bm {U}}) - \mathbf {\Phi }_m^{T}\: \bm {P}= \mbox{$\bf 0$}. \end{aligned} \end{equation}$$ (13)
It should be emphasized that K l i n r e d $\mathbf {K}_{lin \: red}$ as well as the newly established term Φ m T Ω k ( Z T Ω k ) − 1 = : M D E I M ${\mathbf {\Phi }_m^T \, \mathbf {\Omega }_k \, (\mathbf {Z}^T\, \mathbf {\Omega }_k)^{-1} =: \mathbf {M}_{DEIM}}$ are constant throughout the simulation and can be precomputed. Similar to the POD-reduced equation, the residual vector depends on the approximated solution U ¯ $\bar{\bm {U}}$ , but due to the multiplication with Z T $\mathbf {Z}^T$ only the entries of G r e d ( U ¯ ) $\bm {G}_{red}(\bar{\bm {U}})$ corresponding to non-zero rows in the interpolation matrix are of interest. From an implementation standpoint, this multiplication would certainly not be implemented as a multiplication but rather as a query inside the element routine such that only elements containing interpolation DOFs are evaluated. The Newton-Raphson solution scheme for the hyper-reduced problem looks as follows:
( K l i n r e d + M D E I M Z T K T n l ( U ¯ ) Φ m ) Δ U r e d ( i + 1 ) = − G r e d ( U ¯ ( i ) ) U ¯ ( i + 1 ) = U ¯ ( i ) + Φ m Δ U r e d ( i + 1 ) G r e d T ( U ¯ ( i ) ) Δ U r e d ( i + 1 ) ≤ t o l e r a n c e i = i + 1 . $$\begin{equation} \begin{aligned}(\mathbf {K}_{lin \: red}+\mathbf {M}_{DEIM}\: \mathbf {Z}^T\: \mathbf {K}_{T \: nl}(\bar{\bm {U}}) \: \mathbf {\Phi }_m) \: \Delta \bm {U}_{red}^{(i+1)} &= - \bm {G}_{red}(\bar{\bm {U}}{}^{(i)}) \\ \bar{\bm {U}}{}^{(i+1)}&= \bar{\bm {U}}{}^{(i)}+ \mathbf {\Phi }_m\: \Delta \bm {U}_{red}^{(i+1)} \\ \bm {G}_{red}^T(\bar{\bm {U}}{}^{(i)}) \: \Delta \bm {U}_{red}^{(i+1)} &\le tolerance \\ i &= i+1. \end{aligned} \end{equation}$$ (14)
Analogously to the residual vector, only the rows of the stiffness matrix with a corresponding entry in the interpolation matrix are relevant now. Depending on the necessary number of interpolation DOFs, the computational effort to solve the problem can be massively reduced as will be shown in the next section.

5 NUMERICAL EXAMPLE

The considered boundary value problem for the evaluation of the above-described methods is a plate with a circular hole.

In accordance with ref. [8] symmetry conditions are used to simplify the problem such that only a quarter of the plate has to be simulated. It should be noted that this prevents a possible unsymmetrical structural response, which can occur, even though the boundary value problem is symmetric (see e.g. [9]). A visual representation of the problem setup is shown in Figure 1. In addition, the reference points for the displacement and the nonlocal damage are shown in the Figure. The material model used has in total twelve material parameters of which ten are assumed to be known. Two material parameters, namely the first and the second Lamé constants λ and μ are assumed to be uncertain. It is assumed that the first Lamé constant is between 105 and 115 MPa and the second Lamé constant is between 75 and 85 MPa. The question that now arises is, what parameters should be used for the computation of snapshots. As it is beyond the scope of this contribution to go into detail about sampling methods, a simple and intuitive approach is used. The parameter combinations for the precomputations are chosen as the “corners” of the parameter space. The four precomputations are therefore conducted using the parameters { λ , μ } p r e c = { 105 , 75 } , { 105 , 85 } , { 115 , 75 } , { 115 , 85 } $\lbrace \lambda ,\mu \rbrace _{prec} = \lbrace 105,75\rbrace , \lbrace 105, 85\rbrace , \lbrace 115, 75\rbrace , \lbrace 115,85\rbrace$ . To investigate the accuracy of the ROM, its results are compared to those of a full-order model (FOM). The validation parameters for the comparison are chosen as { λ , μ } v a l i = { 110 , 80 } $\lbrace \lambda ,\mu \rbrace _{vali} = \lbrace 110,80\rbrace$ . The load-displacement curves for the simulations with these parameter combinations are shown in Figure 2A. All of the simulations were run for 120 pseudo-timesteps but due to the fact that in real-world applications, the limit load is of most interest, only 80 snapshots of each precomputation were used for the

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FIGURE 1
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General problem setup and problem setup exploiting symmetry conditions.
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FIGURE 2
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Structural response of the plate with a hole.

construction of the ROM. This choice was also made because the authors experienced a decreased accuracy of the ROM when snapshot data from highly nonlinear parts of the simulation was included in the snapshot set even though the latter did not contribute to the structural behavior of the regime that was investigated. For a material behavior as complex as the one used here (elasticity, nonlinear plasticity, nonlinear damage, softening, snapback), it would certainly be beneficial to use clustering methods for the snapshots and create separate ROMs for the identified clusters (see [10]). Furthermore, it would be possible to choose the subspace adaptively (see [11]). In this contribution, however, the default approach is investigated. The damage field after 80 timesteps can be seen in Figure 2B. It should be pointed out here that the damage in point B of the structure is almost 40% which means that the structure is already severely weakened and loaded beyond its limit load. In Figure 3A, the approximation error of the limit load is shown with respect to an increasing number of POD and DEIM modes. The error is defined as ε p m a x = | p m a x F O M − p m a x D E I M | p m a x F O M $\varepsilon _{p_{max}}=\frac{|p_{max \, FOM}-p_{max \, DEIM}|}{p_{max \, FOM}}$ . As expected, a trend can be observed where an increased number of POD and DEIM modes reduce the approximation error up to an order of 10−6. It is, however, also observed that the error cannot be kept consistently low with an increasing number of DEIM modes. The investigation showed that artificial unloading [12] is preventing the structure from being further loaded. Exemplary load-displacement curves for two ROMs with 220 POD and 100 DEIM modes and 260 POD and 40 DEIM modes are shown in Figure 4. The ROM with m = 220 $m=220$ , k = 100 $k=100$ shows that the arc-length method led to undesired unloading after a load of about 35 MPa was reached. The ROM with m = 260 $m=260$ , k = 40 $k=40$ on the other hand was able to reach the desired load state and accurately approximated the response of the structure until after the limit load was reached. The reason for the increased occurrence of the phenomenon in reduced order simulations needs to be further investigated to ensure the stability of the method. The speedup factor of the ROMs is shown in Figure 3B. It is defined as the quotient of the CPU time needed for one iteration of the ROM divided by the time needed for one iteration of the full-order model. The dependence of the speedup factor on the number of POD and DEIM modes can be clearly seen. Looking at the ROM with 260 POD and 40 DEIM modes, the simulation with the ROM was 45 times faster than the simulation using the FOM.

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FIGURE 3
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Approximation error and speedup factor of the ROMs. ROMs, reduced order models.
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FIGURE 4
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Exemplary load-displacement curves of two ROMs. ROMs, reduced order models.

6 CONCLUSION AND OUTLOOK

In this work, it was shown how the POD-based DEIM is applied to structural problems involving complex material behavior. The disadvantage of the POD with regards to nonlinear problems could be successfully addressed by treating specifically the nonlinear part of the equations with the DEIM. The numerical example showed promising results, but also some instabilities that need to be further researched. Mainly the already-known phenomenon of artificial unloading resulted in some ROMs not being able to reach the desired load state. Nevertheless, the simulation was sped up by a significant amount while maintaining an error in the range of 0.0001%. For applications such as uncertainty quantification or optimization problems, the method could therefore massively increase the efficiency. In future works, it shall also be investigated whether the displacement DOFs and the nonlocal damage DOFs can be treated separately from each other. This might reduce the susceptibility to artificial unloading while simultaneously increasing the accuracy of the ROM or achieving the same accuracy with fewer numbers of POD and DEIM modes, leading to a higher speedup factor.

ACKNOWLEDGMENTS

All authors gratefully acknowledge the financial support of the projects TRR339 (subproject B05, project number 453596084) and RE 1057/40 (project number 312911604) by the German Research Foundation (DFG). Further, S. Reese thankfully acknowledges the funding of the project TRR 280 (subproject A01, project number 417002380) by the DFG.

Open access funding enabled and organized by Projekt DEAL.

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