An energy‐stable mixed formulation for isogeometric analysis of incompressible hyperelastodynamics. (25th July 2019)

• Record Type: Journal Article
• Authors: Liu, Ju; Marsden, Alison L.; Tao, Zhen
• Is Part Of: International journal for numerical methods in engineering. Volume 120:Number 8(2019)
• Page Start: 937
• Page End: 963
• Publication Date: 2019-07-25
• Subjects: anisotropic arterial wall model -- energy stability -- generalized‐α method -- incompressible elasticity -- inf‐sup condition -- mixed formulation; Numerical analysis -- Periodicals; Engineering mathematics -- Periodicals; 620.001518
• DOI: 10.1002/nme.6165
• Languages: English
• ISSNs: 0029-5981

Abstract

Summary: We develop a mixed formulation for incompressible hyperelastodynamics based on a continuum modeling framework recently developed in the work of Liu and Marsden and smooth generalizations of the Taylor‐Hood element based on nonuniform rational B‐splines (NURBSs). This continuum formulation draws a link between computational fluid dynamics and computational solid dynamics. This link inspires an energy stability estimate for the spatial discretization, which favorably distinguishes the formulation from the conventional mixed formulations for finite elasticity. The inf‐sup condition is utilized to provide a bound for the pressure field. The generalized‐ α method is applied for temporal discretization, and a nested block preconditioner is invoked for the solution procedure. The inf‐sup stability for different pairs of NURBS elements is elucidated through numerical assessment. The convergence rate of the proposed formulation with various combinations of mixed elements is examined by the manufactured solution method. The numerical scheme is also examined under compressive and tensile loads for isotropic and anisotropic hyperelastic materials. Finally, a suite of dynamic problems is numerically studied to corroborate the stability and conservation properties.