Selective smoothed finite element methods for extremely large deformation of anisotropic incompressible bio‐tissues

Abstract

SUMMARYTwo dynamic selective smoothed FEM (selective S‐FEM) are proposed for analysis of extremely large deformation of anisotropic incompressible bio‐tissues using the simplest four‐node tetrahedron elements. In the present two Selective S‐FEMs, the method that consists of the face‐based smoothed FEM (FS‐FEM) used for the deviatoric part of deformation and the node‐based smoothed FEM (NS‐FEM) used for the volumetric part is called FS/NS‐FEM; another method that replaces the deviatoric part of deformation in the first one by the edge‐based smoothed FEM (3D‐ES‐FEM) is call 3D‐ES/NS‐FEM. Both selective S‐FEMs can achieve outstanding accuracy, and stability of volumetric locking free. This is because the NS‐FEM offers an ‘overly‐soft’ feature (in contrast to the standard FEM ‘overly‐stiff’ model), which can be used to effectively mitigate the volumetric locking, and on the other hand, the 3D‐ES‐FEM and FS‐FEM produce close to exact stiffness for the discretized model leading to accurate solution. Numerical examples are presented to examine the performance of the selective S‐FEM methods, including soft bio‐tissues that may be isotropic, transversely isotropic, and anisotropic arterial layered materials. The present methods are found having good accuracy and performance. The examples also demonstrate that the proposed methods are very robust and possess remarkable capabilities of handling element distortion, which is very useful for simulating soft materials including bio‐tissues. Copyright © 2014 John Wiley & Sons, Ltd.

A cell‐based smoothed finite element method with semi‐implicit CBS procedures for incompressible laminar viscous flows

“…In these S‐FEMs, the ES‐FEM with 3‐node triangular elements and FS‐FEM with 4‐node tetrahedral elements are most preferred in solid mechanics, which is stable and much more accurate and robust than FEM using the same element. The applications of S‐FEM have been seen in many fields, like incompressible hyperelastic solid, large deformation of elastoplastic solids, bio‐tissues, plate and shell, dielectric elastomer, fracture mechanics, fluid‐structure interactions, acoustic‐structure interaction, and electromagnetic field …”

An edge-based/node-based selective smoothed finite element method using tetrahedrons for cardiovascular tissues

“…(43). To verify the implementation of H-G-O model in our in-house codes, three uniaxial tension tests of an adventitial artery strip are conducted by 3D-ES/NS-FEM-T4 [25] using the same setting of reference of Holzapfel et al [47]. The tensile displacement-load curves are agreed with reference curves of Holzapfel et al [47] as shown in Fig.…”

“…Although special efforts [18][19][20] to cure the volumetric locking are also proposed in the frame of FEM using T4 element, it must use the pressure as an extra unknown, in addition to its poor accuracy. Because of these advantages of FS/NS-FEM-T4, it has been applied into the analysis of rubber-like materials [17,21,22], hemodynamics [23], opening of aortal valves [24], and tissues of arterial wall [25]. What is worth to be mentioned is that selective FS/NS-FEM-T4 performs well only for nearly incompressible material and shows slight oscillation of hydrostatic pressure field just like the selective integration technique in FEM.…”

“…It was then further developed for linear static problems [31], structural dynamic problems [32] and fracture mechanics [33,34]. In these previous works, 3D-ES-FEM using T4 element has been found with several interesting merits than other S-FEMs and FEMs, including the nearly-quadratic accuracy and the least time consuming to reach a high accuracy using T4 meshes [24,25]. Hence, applying the 3D-ES-FEM to replace the FS-FEM in Selective S-FEM should develop a more accurate selective S-FEM.…”

F‐bar aided edge‐based smoothed finite element method using tetrahedral elements for finite deformation analysisof nearly incompressible solids

“…There have been already several methods proposed for locking‐free analysis with tetrahedral meshes; yet, they all have some sort of serious drawbacks as follows: Classical u/p hybrid (or mixed) second‐order tetrahedral elements :significant increase in DOF; inevitable Lagrange multipliers; convergence problems in contact analysis; accuracy loss in severely large strain analysis. Other advanced hybrid (or mixed) tetrahedral formulations :significant increase in DOF; inevitable Lagrange multipliers. F‐bar‐Patch method :pressure oscillation in case of too few elements in a patch; low‐energy modes in case of too many elements in a patch; shear locking. Hexahedral elements as subdivisions of tetrahedral elements with B‐bar method and so on (four hexahedra in each tetrahedron) :significant increase in DOF; severe element distortion of initial mesh; pressure oscillation. Appending or blending stabilization methods for tetrahedral nodal integration : need for parameter tuning; limitation of material constitutive model; pressure oscillation. Selective edge/node‐based S‐FEM‐T4 (ES/NS‐FEM‐T4) :limitation of material constitutive model; pressure oscillation; locking at corners. Bubble‐enhanced ES‐FEM‐T4 (bES‐FEM‐T4) :significant increase in DOF; quick pop out of bubble nodes; pressure oscillation. …”