Non‐linear space‐time elasticity

• Published in: International journal for numerical methods in engineering Vol. 124; no. 9; pp. 1965 - 1990
• Main Authors: Schuß, S., Glas, S., Hesch, C.
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 15-05-2023; Wiley Subscription Services, Inc
• Subjects: armijo backtracking; Elasticity; Information transfer; Mathematical analysis; modified Newton method; non‐linear elasticity; polyconvexity; Shape functions; space‐time; Stability; Tetrahedra; upwind stabilization
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.7194

Summary
In this contribution we introduce a novel space‐time formulation for non‐linear elasticity, able to calculate large deformations and displacements with high efficiency using structured and unstructured meshes in the space‐time cylinder without changing the required regularity and thus, enabling the use of Lagrangian shape functions including tetrahedron and hypertetrahedron or tesseract elements. The common and indiscriminate treatment of spatial and temporal directions allows us to remove one of the major bottlenecks in parallel computations: The design of time‐stepping schemes, which can neither been parallelized efficiently in time nor allow for local refinements as no information transfer backwards in time is possible. Moreover, stability of time‐stepping schemes depend on the accumulation of local approximation errors in each time‐step, in contrast to space‐time formulation, characterized by enhanced stability and robustness. Eventually, we will demonstrate superiority in the convergence against classical time‐stepping schemes using various examples including highly sensitive systems in the context of non‐linear elasticity.