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Stability and convergence of in time approximations of hyperbolic elastodynamics via stepwise minimization

Antonín Češík, Sebastian Schwarzacher

TL;DR

The paper develops a two-scale stepwise minimization framework for hyperbolic elastodynamics with large deformations and nonconvex energies. By coupling a velocity-scale $\tau$ and an acceleration-scale $h$, it achieves a variational discretization of the second time derivative and establishes stability via discrete Gronwall-type estimates and a nonconvexity bound for the elastic energy. When the leading-order term is linear, the authors prove convergence with optimal linear rates and, under higher regularity, enhanced space-time regularity results; they also show convergence and rate improvements with a uniformly convex higher-order term. The results are supported by numerical experiments in a simple ODE setting that illustrate rate optimality and the necessity of a small time-step threshold, highlighting the method's potential for large-deformation elastodynamics and related hyperbolic problems.

Abstract

We study step-wise time approximations of non-linear hyperbolic initial value problems. The technique used here is a generalization of the minimizing movements method, using two time-scales: one for velocity, the other (potentially much larger) for acceleration. The main applications are from elastodynamics namely so-called generalized solids, undergoing large deformations. The evolution follows an underlying variational structure exploited by step-wise minimisation. We show for a large family of (elastic) energies that the introduced scheme is stable; allowing for non-linearities of highest order. If the highest order can assumed to be linear, we show that the limit solutions are regular and that the minimizing movements scheme converges with optimal linear rate. Thus this work extends numerical time-step minimization methods to the realm of hyperbolic problems.

Stability and convergence of in time approximations of hyperbolic elastodynamics via stepwise minimization

TL;DR

The paper develops a two-scale stepwise minimization framework for hyperbolic elastodynamics with large deformations and nonconvex energies. By coupling a velocity-scale and an acceleration-scale , it achieves a variational discretization of the second time derivative and establishes stability via discrete Gronwall-type estimates and a nonconvexity bound for the elastic energy. When the leading-order term is linear, the authors prove convergence with optimal linear rates and, under higher regularity, enhanced space-time regularity results; they also show convergence and rate improvements with a uniformly convex higher-order term. The results are supported by numerical experiments in a simple ODE setting that illustrate rate optimality and the necessity of a small time-step threshold, highlighting the method's potential for large-deformation elastodynamics and related hyperbolic problems.

Abstract

We study step-wise time approximations of non-linear hyperbolic initial value problems. The technique used here is a generalization of the minimizing movements method, using two time-scales: one for velocity, the other (potentially much larger) for acceleration. The main applications are from elastodynamics namely so-called generalized solids, undergoing large deformations. The evolution follows an underlying variational structure exploited by step-wise minimisation. We show for a large family of (elastic) energies that the introduced scheme is stable; allowing for non-linearities of highest order. If the highest order can assumed to be linear, we show that the limit solutions are regular and that the minimizing movements scheme converges with optimal linear rate. Thus this work extends numerical time-step minimization methods to the realm of hyperbolic problems.
Paper Structure (19 sections, 35 theorems, 245 equations, 4 figures)

This paper contains 19 sections, 35 theorems, 245 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Stability
  4. General setting for stability
  5. Setting for elastodynamics
  6. Non-convexity estimate for elastic solids
  7. Prototypical energy
  8. Gronwall--type inequalities
  9. The numeric scheme and the definition of stability
  10. A priori bounds on energy
  11. Stability -- elastic solid
  12. Stability -- general
  13. Stability -- ODE
  14. Stability estimate without stabilizing term--elastic solids
  15. Stability estimate without stabilzing term--general
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $E(\eta)=E_1(\eta)+E_2(\eta)$, where $E_1$ is given by kelvinVoigt and $E_2$ by e2:nonlin or e2:lin satisfying eqn:alpha0. Let $\eta_k$ be the variational approximation obtained by step-wise minimization where $\mathcal{E}$ is defined in eq:E and $c>0$. Then there exists a $\tau_0>0$ and a $c_0>0$ depending on the assumptions on $E$, the initial data and the right hand side, such that for all

Figures (4)

  • Figure 1: Comparison of the our approximation $\hat{\eta}_{(\tau)}$ (blue) with the time-delayed solution (dashed) and the limit solution (red). Note that with too large parameter $h=\tau=1$ the solution overshoots and ends up in the other local minimum of $E$.
  • Figure 2: Table showing the predicted error decay of the minimizing movements approximation, for different time steps.
  • Figure :
  • Figure :

Theorems & Definitions (76)

  • Theorem 1.1: Stability for the scheme with artificial viscosity
  • Theorem 1.2: Stability of the direct approach
  • Theorem 1.3: Convergence rate
  • Theorem 1.4: Regularity
  • Remark : Evolutions with dissipation potential
  • Remark : Relevance for existence theory
  • Definition 2.1: Weak solution
  • Definition 2.3
  • Remark : Notation
  • Remark : More general boundary conditions
  • ...and 66 more