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The paper presents a new relatively simple yet very effective method to obtain an approximate solution of the direct and inverse problems for two-dimensional wave equation (two space variables and time). Such a equation describes, for example, the vibration of a membrane. To obtain an approximate solution, the wave polynomials (Trefftz functions for wave equation) were used. It is shown how to get these polynomials and their derivatives. The method of solving the functions is described and it is proved that the approximation error decreases when taking more polynomials in approximation. A new approach for solving 2D direct and inverse problems of elasticity is described. In order to improve the quality of the solution, a physical regularization was proposed. Moreover, the paper shows a new technique of smoothing the noisy data by using wave polynomials. The quality of the approximate solutions was verified on test examples. In these cases the direct and inverse problems were taken into consideration. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Anisotropic meshes are frequently used in many practical problems in order to provide a good resolution of the solutions on the structured mesh and thus to assure the accuracy of the discretization. However, the attributes of anisotropic meshes may have a serious effect on the convergence of linear algebraic solvers. In this paper, we focus mainly on studying the effect of the anisotropic quadrilateral meshes in two dimensions and the anisotropic hexahedral meshes in three dimensions on algebraic multigrid (AMG) solvers. We find that the commonly used AMG methods are less efficient for some locally strong anisotropic meshes. A modified algorithm of coarse grid selection to obtain a more efficient AMG method is then proposed based on the geometric grid information as nodal coordinates. The results of various numerical experiments in linear elasticity are presented. It is shown from the numerical results that the constructed AMG method is robust and efficient not only for meshes containing thin and stretched elements but also for other meshes with locally strong anisotropy. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY Arterial walls in the healthy physiological regime are characterized by quasi-incompressible, anisotropic, hyperelastic material behavior. Polyconvex material functions representing such materials typically incorporate a penalty function to account for the incompressibility. Unfortunately, the penalty will affect the conditioning of the stiffness matrices. For high penalty parameters, the performance of iterative solvers will degrade, and when direct solvers are used, the quality of the solutions will deteriorate. In this paper, an augmented Lagrange approach is used to cope with the quasi-incompressibility condition. Here, the penalty parameter can be chosen much smaller, and as a consequence, the arising linear systems of equations have better properties. An improved convergence is then observed for the finite element tearing and interconnecting-dual primal domain decomposition method, which is used as an iterative solver. Numerical results for an arterial geometry obtained from ultrasound imaging are presented. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper reports the use of polynomial wavelets as approximation functions in hybrid-mixed stress finite element models applied to the solution of plane elasticity problems. The stress and displacement fields in the domain and the displacements on the static boundary are independently approximated. The kinematic boundary conditions are locally satisfied. All remaining equations are enforced in a weighted residual form so designed as to ensure that the discrete model embodies the relevant properties of continuum systems, namely the static-kinematic duality and elastic reciprocity. A set of numerical applications is presented to illustrate the use of the hybrid-mixed model and to assess its efficiency and accuracy. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A clinically applicable approach to estimate the in vivo mechanical material properties of the heart wall is presented. This optimization‐based inverse estimation approach applies a shape‐based objective functional combined with rigid body registration and incremental parameterization of heterogeneity to use standard clinical imaging data along with simplified representations of cardiac function to provide consistent and physically meaningful solution estimates. The capability of the inverse estimation algorithm is evaluated through application to two clinically obtained human datasets to estimate the passive elastic mechanical properties of the heart wall, with an emphasis on the right ventricle. One dataset corresponded to a subject with normal heart function, while the other corresponded to a subject with severe pulmonary hypertension, and therefore expected to have a substantially stiffer right ventricle. Patient‐specific pressure‐driven bi‐ventricle finite element analysis was used as the forward model and the endocardial surface of the right ventricle was used as the target data for the inverse problem. By using the right ventricle alone as the target of the inverse problem the relative sensitivity of the objective function to the right ventricle properties is increased. The method was able to identify material properties to accurately match the corresponding shape of the simplified forward model to the clinically obtained target data, and the properties obtained for the example cases are consistent with the clinical expectation for the right ventricle. Additionally, the material property estimates indicate significant heterogeneity in the heart wall for both subjects, and more so for the subject with pulmonary hypertension. [ABSTRACT FROM AUTHOR]

We test the performance of a three-dimensional finite element, named HC3 , which generalizes the high-continuity (HC) finite element proposed by Aristodemo in 1985 for two-dimensional elastic problems. The HC3 finite element is based on a quadratic B-spline interpolation of the displacement field in three-dimensional linear elasticity. The main feature of this interpolation technique, which can be considered as a particular case of the Bézier interpolation, consists in its capability in reproducing displacement fields of 𝒞1 smoothness with a computational cost equivalent to a linear interpolation, i.e. with a single knot for each element. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The periodontal ligament (PDL) is a soft biological tissue that connects the tooth with the trabecular bone of the mandible. It plays a key role in load transmission and is primarily responsible for bone resorption and most common periodontal diseases. Although several numerical studies have analysed the biomechanical response of the PDL, most did not consider its porous fibrous structure, and only a few analysed damage to the PDL. This study presents an innovative numerical formulation of a porous fibrous hyperelastic damage material model for the PDL. The model considers two separate softening phenomena: fibre alignment during loading and fibre rupture. The parameters for the material model characterization were fitted using experimental data from the literature. Furthermore, the experimental tests used for characterization were computationally modelled to verify the material parameters. A finite element model of a portion of a human mandible, obtained by microcomputerized tomography, was developed, and the proposed constitutive model was implemented for the PDL. Our results confirm that damage to the PDL may occur mainly because of overpressure of the interstitial fluid, while large forces must be applied to damage the PDL fibrous network. Moreover, this study clarifies some aspects of the relationship between PDL damage and the bone remodelling process. This study considers separately two softening phenomena of the periodontal ligament (PDL): the first related to collagen fibres alignment and the second to fibres rupture. This behaviour was formulated by a porous fibrous hyperelastic damage material model. Its material parameters were obtained mimicking experimental tests of the literature. Afterwards, the material behaviour was implemented in a finite element model of a portion of a human mandible to analyse the PDL mechanical response under different loading conditions. [ABSTRACT FROM AUTHOR]