Summary A mean‐strain 10‐node tetrahedral element is developed for the solution of geometrically nonlinear solid mechanics problems using the concept of energy‐sampling stabilization. A uniform‐strain tetrahedron for applications to linear elasticity was recently described. The formulation as extended here is able to solve large‐strain hyperelasticity. The present 10‐node tetrahedron is composed of several four‐node linear tetrahedral elements, four tetrahedra in the corners, and four tetrahedra that tile the central octahedron in three possible sets of four‐node tetrahedra, corresponding to three different choices for the internal diagonal. We formulate a mean‐strain element with stabilization energy evaluated on the four corner tetrahedra, which is shown to guarantee consistency and stability. The stabilization energy is expressed through a stored‐energy function, and contact with input parameters in the small‐strain regime is made. The neo‐Hookean model is used to formulate the stabilization energy. As for small‐strain elasticity, the stabilization parameters are determined by actual material properties and geometry of a tetrahedra without any user input. The numerical tests demonstrate that the present element performs well for solid, shell, and nearly incompressible structures. Copyright © 2016 John Wiley & Sons, Ltd.

Summary In this study, a new mean‐strain 10‐node tetrahedral element is developed using energy‐sampling stabilization. The proposed 10‐node tetrahedron is composed of several four‐node linear tetrahedral elements, four tetrahedra in the corners and four tetrahedra that tile the central octahedron in three possible sets of four‐node linear tetrahedra, corresponding to three different choices for the internal diagonal. The assumed strains are calculated from mean ‘basis function gradients.’ The energy‐sampling technique introduced previously for removing zero‐energy modes in the mean‐strain hexahedron is adapted for the present element: the stabilization energy is evaluated on the four‐corner tetrahedra. The proposed element naturally leads to a lumped‐mass matrix and does not have unphysical low‐energy vibration modes. For simplicity, we limit our developments to linear elasticity with compressible and nearly incompressible material. The numerical tests demonstrate that the present element performs well compared with the classical 10‐node tetrahedral elements for shell and plate structures, and nearly incompressible materials. Copyright © 2016 John Wiley & Sons, Ltd.

Summary The concept of energy‐sampling stabilization is used to develop a mean‐strain quadratic 10‐node tetrahedral element for the solution of geometrically nonlinear solid mechanics problems. The development parallels recent developments of a “composite” uniform‐strain 10‐node tetrahedron for applications to linear elasticity and nonlinear deformation. The technique relies on stabilization by energy sampling with a mean‐strain quadrature and proposes to choose the stabilization parameters as a quasi‐optimal solution to a set of linear elastic benchmark problems. The accuracy and convergence characteristics of the present formulation are tested on linear and nonlinear benchmarks and compare favorably with the capabilities of other mean‐strain and high‐performance tetrahedral and hexahedral elements for solids, thin‐walled structures (shells), and nearly incompressible structures.

This paper presents a new formulation for forward scalar wave simulations in semi-infinite media. Perfectly-Matched-Layers (PMLs) are used as a wave absorbing boundary layer to surround a finite computational domain truncated from the semi-infinite domain. In this work, a hybrid formulation was developed for the simulation of scalar wave motion in two-dimensional PML-truncated domains. In this formulation, displacements and stresses are considered as unknowns in the PML domain, while only displacements are considered to be unknowns in the interior domain. This formulation reduces computational cost compared to fully-mixed formulations. To obtain governing wave equations in the PML region, complex coordinate stretching transformation was introduced to equilibrium, constitutive, and compatibility equations in the frequency domain. Then, equations were converted back to the time-domain using the inverse Fourier transform. The resulting equations are mixed (contain both displacements and stresses), and are coupled with the displacement-only equation in the regular domain. The Newmark method was used for the time integration of the semi-discrete equations.

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