Local stabilized isogeometric method solves nearly incompressible elasticity issues
A Local Stabilized Isogeometric Formulation with Discontinuous Pressure for Mesh-Conforming and Immersed Analysis of Nearly Incompressible Elasticity
Computational Engineering, Finance, and Science
Summary
Modeling very soft materials like gels and tissues on computers is hard because standard methods get stuck with inaccuracies when materials barely change volume. The authors developed a way to mix different mathematical quantities that keeps calculations stable, efficient, and accurate even for difficult materials. Their method works well with both detailed meshes and scenarios where the shape isn’t perfectly lined up with the computational grid. This helps simulate soft materials more reliably in engineering and medical applications.
What this means in practice
- •For finite element software developers: Implement a stable mixed formulation with local pressure condensation for efficient simulation of nearly incompressible soft materials.
- •For biomechanical engineers: Simulate soft tissue mechanics accurately using immersed methods that handle complex geometries without mesh conformity.
Authors
Ashkan Ali Madadi, Thomas J. R. Hughes
Abstract
Nearly incompressible materials arise in soft tissues, elastomers, gels, and other soft solids. Yet, numerical modeling remains challenging because displacement-based formulations can exhibit volumetric locking near incompressibility. Mixed displacement-pressure formulations address this difficulty but require compatible approximation spaces satisfying the inf-sup condition. In mixed formulations, discontinuous pressure spaces are attractive since pressure degrees of freedom can be condensed locally. In high-order isogeometric analysis, however, their Galerkin stability requires pressure-grid coarsening that increases with polynomial degree. Here, we develop a variational multiscale (VMS) stabilized mixed isogeometric formulation that retains this locality while substantially relaxing the restriction. The key observation is that VMS stabilization controls nonconstant elementwise pressure modes and is unconditionally stable with respect to its parameter, provided the elementwise constant pressure component satisfies the inf-sup condition. Consequently, for maximally continuous displacement splines, a pressure grid coarsened by a factor of two satisfies the constant-mode inf-sup criterion independently of polynomial degree. Local condensation then yields a displacement-only system, facilitating use in existing displacement solvers. The formulation is further extended to immersed isogeometric analysis with extended B-splines, pressure-cell aggregation, subdivision-based cut-cell quadrature, and weak enforcement of essential boundary conditions. Discrete inf-sup studies and convergence analyses in the Kirsch and Cook's membrane benchmarks demonstrate stability and show that the immersed treatment preserves mixed stability properties. The proposed framework therefore provides a local, high-order, and robust approach for nearly incompressible problems in mesh-conforming and immersed settings.