A Robust Solution Procedure for Hyperelastic Solids with Large Boundary Deformation

Compressible Mooney-Rivlin theory has been used to model hyperelastic solids, such as rubber and porous polymers, and more recently for the modeling of soft tissues for biomedical tissues, undergoing large elastic deformations. We propose a solution procedure for Lagrangian finite element discretization of a static nonlinear compressible Mooney-Rivlin hyperelastic solid. We consider the case in which the boundary condition is a large prescribed deformation, so that mesh tangling becomes an obstacle for straightforward algorithms. Our solution procedure involves a largely geometric procedure to untangle the mesh: solution of a sequence of linear systems to obtain initial guesses for interior nodal positions for which no element is inverted. After the mesh is untangled, we take Newton iterations to converge to a mechanical equilibrium. The Newton iterations are safeguarded by a line search similar to one used in optimization. Our computational results indicate that the algorithm is up to 70 times faster than a straightforward Newton continuation procedure and is also more robust (i.e., able to tolerate much larger deformations). For a few extremely large deformations, the deformed mesh could only be computed through the use of an expensive Newton continuation method while using a tight convergence tolerance and taking very small steps.Comment: Revision of earlier version of paper. Submitted for publication in Engineering with Computers on 9 September 2010. Accepted for publication on 20 May 2011. Published online 11 June 2011. The final publication is available at http://www.springerlink.co

Distortion and quality measures for validating and generating high-order tetrahedral meshes

A procedure to quantify the distortion (quality) of a high-order mesh composed of curved tetrahedral elements is presented. The proposed technique has two main applications. First, it can be used to check the validity and quality of a high-order tetrahedral mesh. Second, it allows the generation of curved meshes composed of valid and high-quality high-order tetrahedral elements. To this end, we describe a method to smooth and untangle high-order tetrahedral meshes simultaneously by minimizing the proposed mesh distortion. Moreover, we present a -continuation procedure to improve the initial configuration of a high-order mesh for the optimization process. Finally, we present several results to illustrate the two main applications of the proposed technique.Peer ReviewedPostprint (author’s final draft

A distortion measure to validate and generate curved high-order meshes on CAD surfaces with independence of parameterization

This is the accepted version of the following article: [Gargallo-Peiró, A., Roca, X., Peraire, J., and Sarrate, J. (2016) A distortion measure to validate and generate curved high-order meshes on CAD surfaces with independence of parameterization. Int. J. Numer. Meth. Engng, 106: 1100–1130. doi: 10.1002/nme.5162], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5162/abstractA framework to validate and generate curved nodal high-order meshes on Computer-Aided Design (CAD) surfaces is presented. The proposed framework is of major interest to generate meshes suitable for thin-shell and 3D finite element analysis with unstructured high-order methods. First, we define a distortion (quality) measure for high-order meshes on parameterized surfaces that we prove to be independent of the surface parameterization. Second, we derive a smoothing and untangling procedure based on the minimization of a regularization of the proposed distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes to enforce that the nodes slide on the surfaces. Moreover, the proposed algorithm repairs invalid curved meshes (untangling), deals with arbitrary polynomial degrees (high-order), and handles with low-quality CAD parameterizations (independence of parameterization). Third, we use the optimization procedure to generate curved nodal high-order surface meshes by means of an a posteriori approach. Given a linear mesh, we increase the polynomial degree of the elements, curve them to match the geometry, and optimize the location of the nodes to ensure mesh validity. Finally, we present several examples to demonstrate the features of the optimization procedure, and to illustrate the surface mesh generation process.Peer ReviewedPostprint (author's final draft

Addressing some current issues in linear and high-order meshing

The thesis explores the generation of anisotropic and boundary conforming Voronoi regions and Delaunay triangulations, high-order mesh quality and the development of mesh enhancement techniques which incorporate quality measures to preserve mesh validity longer. In the first part an analogy with crystal growth is proposed to handle mesh anisotropy and boundary conformity in Voronoi diagrams and Delaunay mesh generation. A Voronoi partition of a domain corresponds to the steady-state configuration of many crystals growing from their seed points. Mesh anisotropy is incorporated and the shape of the boundary of an isolated crystal is guided by re-interpreting a user-defined Riemann metric in terms of the velocity of the crystal boundary. A straightforward implementation of conformity to boundaries is achieved by treating the boundary of the computational domain as the boundary of a stationary crystal. The second part attempts to answer the question: what is a good highorder element? A review of a priori mesh quality measures suitable for high-order elements is presented. A systematic analysis of the quality measures for interior and boundary elements is then carried out utilising a number of test cases that consist of a set of carefully selected elements with various degrees of distortion. Their ability to identify severe geometrical distortion is discussed. The effect of boundary curvature on the performance of quality measures is also investigated. The last part proposes improvements to a conventional mesh deformation method based on the equations of elasticity to maintain highorder mesh validity and enhance mesh quality. This is accomplished by incorporating additional terms, that can be interpreted as body forces and thermal stresses in the elastic analogy. Different test cases are designed to prolong mesh validity, and their performance is reported. A proposal of how to formulate these terms to incorporate anisotropy is also presented.Open Acces

Validation and generation of curved meshes for high-order unstructured methods

In this thesis, a new framework to validate and generate curved high-order meshes for complex models is proposed. The main application of the proposed framework is to generate curved meshes that are suitable for finite element analysis with unstructured high-order methods. Note that the lack of a robust and automatic curved mesh generator is one of the main issues that has hampered the adoption of high-order methods in industry. Specifically, without curved high-order meshes composed by valid elements and that match the domain boundary, the convergence rates and accuracy of high-order methods cannot be realized. The main motivation of this work is to propose a framework to address this issue. First, we propose a definition of distortion (quality) measure for curved meshes of any polynomial degree. The presented measures allow validating if a high-order mesh is suitable to perform finite element analysis with an unstructured high-order method. In particular, given a high-order element, the measures assign zero quality if the element is invalid, and one if the element corresponds to the selected ideal configuration (desired shape and nodal distribution). Moreover, we prove that if the quality of an element is not zero, the region where the determinant of the Jacobian is not positive has measure zero. We present several examples to illustrate that the proposed measures can be used to validate high-order isotropic and boundary layer meshes. Second, we develop a smoothing and untangling procedure to improve the quality for curved high-order meshes. Specifically, we propose a global non-linear least squares minimization of the defined distortion measures. The distortion is regularized to allow untangling invalid meshes, and it ensures that if the initial configuration is valid, it never becomes invalid. Moreover, the optimization procedure preserves, whenever is possible, some geometrical features of the linear mesh such as the shape, stretching, straight-sided edges, and element size. We demonstrate through examples that the implementation of the optimization problem is robust and capable of handling situations in which the mesh before optimization contains a large number of invalid elements. We consider cases with polynomial approximations up to degree ten, large deformations of the curved boundaries, concave boundaries, and highly stretched boundary layer elements. Third, we extend the definition of distortion and quality measures to curved high-order meshes with the nodes on parameterized surfaces. Using this definition, we also propose a smoothing and untangling procedure for meshes on CAD surfaces. This procedure is posed in terms of the parametric coordinates of the mesh nodes to enforce that the nodes are on the CAD geometry. In addition, we prove that the procedure is independent of the surface parameterization. Thus, it can optimize meshes on CAD surfaces defined by low-quality parameterizations. Finally, we propose a new mesh generation procedure by means of an a posteriori approach. The approach consists of modifying an initial linear mesh by first, introducing high-order nodes, second, displacing the boundary nodes to ensure that they are on the CAD surface, and third, smoothing and untangling the resulting mesh to produce a valid curved high-order mesh. To conclude, we include several examples to demonstrate that the generated meshes are suitable to perform finite element analysis with unstructured high-order methods.Postprint (published version