Narrow Results

This paper further analyzes three Bathe algorithms ($\gamma$-Bathe, ${\beta }_1∕{\beta }_2$-Bathe and ${\rho }_{\infty }$-Bathe) with their unknown properties revealed. The analysis shows firstly that three Bathe algorithms can cover two common integration schemes, trapezoidal rule and backward Euler formula, and that the second-order ${\beta }_1∕{\beta }_2$-Bathe algorithm is algebraically identical to the ${\rho }_{\infty }$-Bathe algorithm. Via formulation of the generalized two-sub-step Newmark algorithm, it is shown that the common Newmark method cannot be considered as a special case of the ${\rho }_{\infty }$-Bathe algorithm. For wave propagation problems, optimal Courant–Friedrichs–Lewy (CFL) numbers for reducing dispersion errors are found for the three Bathe algorithms by considering spatial and temporal discretizations simultaneously, while the modified integration rules are used for the element mass and stiffness matrices to reduce the anisotropy in wave propagating directions. The recommended optimal algorithmic parameters are given for the three Bathe algorithms to help users effectively solve various dynamic and wave propagation problems.

An edge-based smoothed tetrahedron radial point interpolation method (ES-T-RPIM) is formulated for the 3D acoustic problems, using the simplest tetrahedron mesh which is adaptive for any complicated geometry. In present ES-T-RPIM, the gradient smoothing operation is performed with respect to each edge-based smoothing domain, which is also serving as building blocks in the assembly of the stiffness matrix. The smoothed Galerkin weak form is then used to create the discretized system equations. The acoustic pressure is constructed using radial point interpolation method, and two typical schemes of selecting nodes for interpolation using RPIM have been introduced in detail. It turns out that the ES-T-RPIM provides an ideal amount of softening effect, and significantly reduces the numerical dispersion error in low- to mid-frequency range. Numerical examples demonstrate the superiority of the ES-T-RPIM for 3D acoustic analysis, especially at mid-frequency.

Due to the “overly-stiff” nature, it has been proved that the finite element method (FEM) is inevitable to suffer from the dispersion error when solving mid-frequency acoustic problems. In order to overcome this defect, in this study, a novel virtual node selection scheme of changing the virtual node’s position with a parameter is proposed, which is used to construct the condensed shape function of the cell-based radial point interpolation method (CS-RPIM) to soften the stiffness of discrete model. Based on the generalized gradient smoothing technique (GGST) and the condensed shape function technique, the present method provides a “close-to-exact” stiffness for discrete system model, which controls the dispersion error from the root. Several numerical examples have been researched to verify the performance of the proposed method in mid-frequency acoustic simulations. Moreover, with the excellent robustness, the present method lays a good foundation for solving complex practical engineering problems.

Elastography is an imaging approach to measure the stiffness of tissues to provide diagnostic information. Currently, finite element method (FEM) has been widely used in elastography. However, FEM tends to an overly stiff model that sometimes gives unsatisfactory accuracy, particularly using triangular elements in 2D or tetrahedral elements in 3D. In general, it is difficult or even impossible to generate quadrilateral or brick elements to precisely capture the anatomic details for mechanobiologic modeling as the biologic system can be rather sophisticated. In addition, biologic soft tissues are often considered as “incompressible” materials, where conventional FEM could suffer from volumetric locking in numerical solution. On the other hand, linear triangular and tetrahedral mesh can be automatically generated for complicated geometry, which significantly saves the time for the creation of model. With these reasons, for the first time, smoothed finite element method (SFEM) is developed to analyze elastography problems. A range of numerical examples, including static, dynamic, viscoelastic and time harmonic cases have exemplified herein to validate that SFEM is able to provide more accurate and stable solutions using the same set of mesh compared with the standard FEM. Furthermore, SFEM is also effective to inversely compute the mechanical properties of abnormal tissue.