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A stable DG-POD reduced-order method for dynamic linear viscoelasticity
Byeong-Chun Shin, Yongseok Jang
TL;DR
Dynamic viscoelastic simulations remain costly because hereditary memory and large DG systems burden long-time computation. This paper develops a stable DG-POD framework using internal variables and displacement-based reduction, with theory and experiments showing accurate reduced dynamics and substantial computational savings.
Problem
Hereditary memory and large discontinuous Galerkin systems make dynamic viscoelastic simulations computationally intensive, particularly for long-time and many-query problems.
Method
The framework combines internal-variable reformulation, symmetric interior-penalty DG discretization, and POD projection of the displacement system.
Results
The method preserves DG convergence and dominant full-order dynamics while reducing computational cost, with approximately 7.5× overall speed-up including snapshot generation and basis construction.
Takeaways & Limitations
DG-POD supports accurate, efficient long-time simulation of dynamic linear viscoelasticity, while richer reduced spaces improve resolution of finer transient oscillations.
Takeaways & Limitations
The analysis assumes a positive-definite elasticity tensor with standard symmetries and a quasi-uniform spatial partition.
Abstract
from arXiv · showhide
We present a reduced-order modeling framework based on proper orthogonal decomposition for dynamic linear viscoelasticity governed by generalized Maxwell models. The hereditary constitutive law is reformulated using internal variables, and the resulting problem is discretized in space by a symmetric interior penalty discontinuous Galerkin method and in time by either the backward Euler or Crank--Nicolson scheme. The reduced model is obtained by projecting the fully discrete system onto low-dimensional spaces constructed from displacement snapshots. For the full-order discontinuous Galerkin formulation, we show well-posedness and derive an a priori stability bound without using Gronwall's inequality. Consequently, the bound grows at most linearly, rather than exponentially, with the final time. We further derive spatial error estimates and an error decomposition that separates the effects of spatial discretization, temporal discretization, and proper orthogonal decomposition truncation. The reduced formulation replaces the large full-order systems with low-dimensional problems, enabling efficient long-time simulations. Numerical experiments verify the predicted convergence behavior, assess different inner products for constructing the reduced basis, and demonstrate accurate recovery of transient oscillations and viscoelastic relaxation with substantial computational savings, even after accounting for snapshot generation and basis construction.
1. Introduction
Dynamic viscoelastic simulations are costly because hereditary constitutive laws require solution-history storage and repeated evaluation, while DG discretizations add substantial degrees of freedom. This work develops a POD reduced-order framework combining internal variables, displacement-based projection, and symmetric interior penalty DG, and examines how POD inner products affect accuracy.
- Motivation: Hereditary constitutive laws create significant computational overhead by requiring storage and repeated evaluation of the complete solution history.Internal-variable reformulations reduce this burden for exponential relaxation kernels, but full-order systems remain large.
- Motivation: DG methods offer local conservation, geometric flexibility, high-order accuracy, hp-adaptivity, and parallel scalability for transient wave propagation problems.Their increased number of degrees of freedom nevertheless contributes to computational cost in dynamic viscoelasticity.
- Reduced-order modeling: POD constructs low-dimensional approximation spaces from representative solution snapshots and provides a simple, robust, and computationally efficient reduction approach for time-dependent partial differential equations.The method of snapshots offers a practical framework for computing POD bases from data.
- Contributions: The proposed method develops a POD reduced-order model for DG discretizations of second-order viscoelastic wave equations using internal-variable formulations and projection-based model reduction.The work also investigates how different POD inner products influence reduced-model accuracy within the DG framework.
- Novelty: The framework combines internal-variable reformulation, displacement-based solution, and POD within a symmetric interior penalty DG discretization of dynamic linear viscoelasticity.Internal variables replace hereditary convolution with local evolution equations, avoiding repeated quadrature over the complete deformation history.
2. Model problem
The model describes dynamic linear viscoelasticity through momentum balance, boundary and initial conditions, and a hereditary constitutive law. A Prony-series generalized Maxwell model is reformulated with internal variables, replacing history dependence by a local-in-time coupled system.
- Governing problem: The problem concerns displacement and stress evolution in a bounded, homogeneous, isotropic viscoelastic solid over a prescribed time interval.The domain is polytopic, and the material has constant mass density.
- Governing problem: The boundary is split into disjoint Dirichlet and Neumann parts, with homogeneous displacement conditions, prescribed tractions, and specified initial data.The Dirichlet boundary is assumed to have positive surface measure.
- Constitutive relation: The constitutive law uses a Volterra hereditary integral in which a positive-definite elasticity tensor and relaxation function link stress to deformation history.The elasticity tensor satisfies the standard minor and major symmetries, while the relaxation function characterizes material stress relaxation.
- Internal-variable reformulation: For the generalized Maxwell solid, the relaxation function is represented by a Prony series with internal variables associated with individual relaxation mechanisms.Each internal variable is defined through a temporal convolution and represents one relaxation mode’s contribution.
- Internal-variable reformulation: The internal-variable formulation replaces the nonlocal Volterra history dependence with coupled local-in-time displacement and evolution equations suitable for numerical discretization.The adopted velocity form is mathematically equivalent to an alternative displacement form and is often more convenient computationally.
3. Discontinuous Galerkin method
The section constructs the SIPG discontinuous Galerkin discretization for viscoelasticity and establishes the algebraic properties, well-posedness, stability, and approximation foundations of the semi-discrete method.
- Mesh and finite element spaces: The method partitions Ω into triangular or tetrahedral elements, assumes a quasi-uniform mesh, and defines broken Sobolev and discontinuous polynomial spaces.Interior and boundary faces are equipped with prescribed unit normals, while jumps, averages, and penalty operators support the DG formulation.
- DG bilinear form: For sufficiently large penalty parameters with β0(d−1) ≥1, the SIPG bilinear form is coercive and continuous in the DG energy norm.The result follows from inverse polynomial trace estimates and bounds on interior penalty terms.
- Algebraic formulation: The semi-discrete formulation yields a matrix ODE system whose mass matrix M and SIPG stiffness matrix A are symmetric positive definite, while jump matrix J is symmetric positive semidefinite.These properties require α0 > 0 sufficiently large and β0(d−1) ≥1.
- Well-posedness: For continuous forcing and prescribed initial data with z_q(0) = 0, the semi-discrete system admits a unique solution on [0, T].Invertibility of M converts the equations into a first-order system with a globally Lipschitz right-hand side.
- Approximation error: Under coercivity and sufficient solution regularity, the analysis provides a semi-discrete DG approximation estimate with a constant independent of h.The section introduces this estimate after establishing the stability bound and auxiliary DG inequalities.
- Stability: Avoiding Grönwall’s inequality produces a stability bound that grows at most linearly, rather than exponentially, with final time T, supporting long-time simulations.The stability theorem assumes sufficiently large α0 and β0(d−1) ≥1 so the SIPG bilinear form is coercive.
4. Proper orthogonal decomposition
This section develops a DG-POD reduced-order model from displacement snapshots, using optimal or weighted POD bases and Galerkin projection. It also describes efficient time stepping and decomposes the total error into spatial, temporal, and POD truncation contributions.
- POD framework: The DG-POD framework reduces computational complexity by projecting the semi-discrete DG system onto a low-dimensional POD space.The POD basis is formed from dominant snapshot information and used to derive the reduced-order model.
- POD basis construction: The POD basis consists of orthonormal modes spanning the dominant snapshot subspace and minimizes mean-square projection error among all 𝓁-dimensional subspaces.It is constructed from the first 𝓁 left singular vectors, equivalently using the dominant eigenvectors of the snapshot correlation matrix.
- Weighted POD: Weighted POD generalizes the standard inner product through a symmetric positive definite weight, including natural mass- or energy-based choices.The weighted basis can be computed from the weighted snapshot SVD or the weighted correlation matrix.
- Reduced simulation: The implementation builds the POD space from displacement snapshots, reuses it for velocity and internal variables, and solves a reduced linear system at each time step.The remaining reduced variables are updated algebraically, and the full DG finite element solution is reconstructed afterward.
- Error decomposition: The total error separates spatial discretization, temporal discretization, and POD truncation, with time orders 𝑂(Δ𝑡^1) for backward Euler and 𝑂(Δ𝑡^2) for Crank-Nicolson.The spatial DG approximation has energy-norm order 𝑂(ℎ^min(𝑘+1,𝑠)−1), while reduced-order error is controlled by the POD projection error under uniform reduced stability.
- Computational efficiency: Time-marching complexity depends on the reduced dimension 𝓁 rather than the full-order dimension 𝑛dof, supporting efficient long-time simulations.The associated stability bound grows at most linearly with final time rather than exponentially.
5. Numerical experiments
Numerical experiments show that DG-POD retains the expected spatial and temporal convergence behavior while substantially reducing time-stepping costs. Increasing the reduced basis improves resolution of oscillatory transients, with strong speed-ups remaining after snapshot-generation costs.
- Spatial convergence: DG-POD achieves optimal spatial convergence rates, O(h^k+1) in L2 and O(h^k) in the broken H1 norm, matching full-order DG behavior.For k = 2, only the finest-mesh L2 result with W_u = M shows a slight deterioration.
- Computational performance: DG-POD significantly reduces time-stepping cost, with solve-time speed-up increasing under spatial refinement and approximately 80 for both temporal schemes.Combined set-up and solve speed-up increases with the number of time steps, while total speed-up is smaller when snapshot generation is included.
- Temporal convergence: Backward Euler exhibits first-order temporal convergence, while Crank–Nicolson achieves second-order convergence in the maximum-in-time L2 norm.The three POD inner products yield nearly identical temporal accuracy.
- Reduced basis dimension: Increasing the POD dimension from l = 10 to l = 200 reduces the relative POD tail from 1.46 × 10−1 to 6.52 × 10−3 and recovers finer oscillatory features.The highly oscillatory transient response requires additional modes to reduce truncation error.
- Reduced basis dimension: At l = 200, DG-POD is approximately 5.19 × 103 times faster in the solving phase, while total time is approximately 78s versus 582s for DG-FOM.Larger reduced dimensions increase reduced-system setup and solution costs, but substantial savings remain after snapshot generation.
6. Conclusion
The paper establishes a stable DG-POD framework for dynamic linear viscoelasticity, with well-posedness, spatial error estimates, and an error decomposition for fully discrete approximations. Numerical experiments confirm accurate, robust, and computationally efficient reduced simulations that preserve the full-order model’s dominant dynamics and convergence behavior.
- Contributions: The proposed DG-POD method uses an internal-variable formulation for dynamic linear viscoelasticity.The analysis covers both the semi-discrete DG problem and the fully discrete DG-POD approximation.
- Contributions: The study establishes well-posedness and derives a priori spatial error estimates for the semi-discrete DG problem.
- Contributions: The fully discrete approximation separates spatial discretization, temporal discretization, and POD truncation contributions through an error decomposition.
- Numerical findings: Numerical experiments confirm accuracy, robustness, and computational efficiency across smooth benchmark problems and transient wave propagation.The reduced model reproduces the dominant dynamics of the full-order system at lower computational cost.
- Numerical findings: The reduced model preserves the convergence behavior of the underlying DG discretization while revealing an accuracy–complexity tradeoff in reduced-space dimension.Low-dimensional reduced spaces capture the large-scale dynamics with reduced computational cost.
CRediT authorship contribution statement
The authors contributed across investigation, methodology, analysis, software, visualization, and manuscript preparation.
- Byeong-Chun Shin contributed investigation, methodology, and original-draft writing, review, and editing.
- Yongseok Jang contributed formal analysis, investigation, methodology, software, visualization, and original-draft writing, review, and editing.