Fractional-step finite difference schemes for incompressible elasticity on overset grids

• Published in: Journal of computational physics 2023-09, Vol.488, p.112221, Article 112221
• Main Authors: Banks, J.W., Henshaw, W.D., Newell, A., Schwendeman, D.W.
• Format: Article
• Language: English
• Subjects: Fractional-step methods, overset grids; Incompressible elasticity
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2023.112221

Efficient finite-difference schemes for the numerical solution of the time-dependent equations of incompressible linear elasticity on complex geometry are presented. The schemes are second-order accurate in space and time, and solve the equations in displacement-pressure form. The algorithms use a fractional-step approach in which the time-step of the displacements is performed separately from the solution of a Poisson problem to update the pressure. Complex geometry is treated with curvilinear overset grids. Compatibility boundary conditions and an upwind dissipation for wave equations in second-order form are included to ensure stability for overset grids, and for the difficult case of traction (free-surface) boundary conditions. A divergence damping term is added to keep the dilatation small. The stability of the schemes is studied with GKS mode analysis. Exact eigen-mode solutions are obtained for several problems involving rectangular, cylindrical and spherical configurations, and these are valuable as benchmark problems. The accuracy and stability of the approach in two and three space dimensions is illustrated by comparing the numerical solutions with the exact solutions of the benchmark problems. •Stable finite difference schemes for incompressible elasticity in second-order form.•Fractional-step schemes for displacement and pressure with divergence damping.•Compatibility boundary conditions for second-order accurate fractional-step schemes.•Upwind schemes for incompressible elasticity on overset (overlapping) grids.•Exact solutions of benchmark problems for incompressible vibrational modes.