Polyconvexity inspired frameworks and structure‐preserving integrators for multi‐field problems. Issue 1 (17th December 2018)

• Record Type: Journal Article
• Authors: Franke, Marlon; Janz, Alexander; Schiebl, Mark; Ortigosa, Rogelio; Betsch, Peter
• Other Names: Müller G. guestEditor.; Ulbrich M. guestEditor.
• Is Part Of: Proceedings in applied mathematics and mechanics. Volume 18:Issue 1(2018)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2018-12-17
• Subjects: Applied mathematics -- Periodicals; Engineering mathematics -- Periodicals; Mathematical physics -- Periodicals; 519
• DOI: 10.1002/pamm.201800010
• Languages: English
• ISSNs: 1617-7061

Abstract

Abstract: A new approach for the design of energy‐momentum (EM) consistent time integrators for nonlinear coupled problems is proposed. Polyconvexity inspired internal or Helmholtz free energy functions are obtained by using the rediscovered tensor cross product (see [1]) which is basically applied on the cofactor and the determinant of the right Cauchy‐Green strain tensor and greatly simplifies the algebra (see [1, 2]). On this basis multi‐field problems concerning non‐linear thermo‐elastodynamics (see [3]) and electro‐elastodynamics (see [4]) are considered. For the former a temperature based weak form is employed which facilitates the design of a structure‐preserving time‐stepping scheme for coupled thermo‐elastic problems. For the latter a four‐field internal energy‐based dynamic formulation is applied. In both cases, the polyconvexity‐based framework facilitates the design of EM consistent time integrators. In particular algorithmic stress formulas are employed which show a remarkably simple structure when compared to traditional, elaborate projection‐based formulas. The spatial discretisation relies on finite element interpolations for the unknown fields. Eventually, the superior robustness of the proposed formulations is shown in two numerical examples.