Geometric diagram for representing shape quality in mesh refinement

summary:We review and discuss a method to normalize triangles by the longest-edge. A geometric diagram is described as a helpful tool for studying and interpreting the quality of triangle shapes during iterative mesh refinements. Modern CAE systems as those implementing the finite element method (FEM) require such tools for guiding the user about the quality of generated triangulations. In this paper we show that a similar method and corresponding geometric diagram in the three-dimensional case do not exist

Analysis of and workarounds for element reversal for a finite element-based algorithm for warping triangular and tetrahedral meshes

We consider an algorithm called FEMWARP for warping triangular and tetrahedral finite element meshes that computes the warping using the finite element method itself. The algorithm takes as input a two- or three-dimensional domain defined by a boundary mesh (segments in one dimension or triangles in two dimensions) that has a volume mesh (triangles in two dimensions or tetrahedra in three dimensions) in its interior. It also takes as input a prescribed movement of the boundary mesh. It computes as output updated positions of the vertices of the volume mesh. The first step of the algorithm is to determine from the initial mesh a set of local weights for each interior vertex that describes each interior vertex in terms of the positions of its neighbors. These weights are computed using a finite element stiffness matrix. After a boundary transformation is applied, a linear system of equations based upon the weights is solved to determine the final positions of the interior vertices. The FEMWARP algorithm has been considered in the previous literature (e.g., in a 2001 paper by Baker). FEMWARP has been succesful in computing deformed meshes for certain applications. However, sometimes FEMWARP reverses elements; this is our main concern in this paper. We analyze the causes for this undesirable behavior and propose several techniques to make the method more robust against reversals. The most successful of the proposed methods includes combining FEMWARP with an optimization-based untangler.Comment: Revision of earlier version of paper. Submitted for publication in BIT Numerical Mathematics on 27 April 2010. Accepted for publication on 7 September 2010. Published online on 9 October 2010. The final publication is available at http://www.springerlink.co

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

Adaptation de maillages hybrides et application aux simulations d'équipements de combustion.

RÉSUMÉ Dans le domaine de la simulation des ´ecoulements r´eactifs en milieu industriel, sp´ecifiquement pour la conception des ´equipements de combustion pour les chaudi`eres industrielles, les ressources de calculs sont limit´ees et la dur´ee du cycle de conception est tr`es courte. La taille des domaines simul´es est de l’ordre de plusieurs ´echelles audessus de la taille des ´el´ements n´ecessaires pour capturer pr´ecis´ement les ph´enom`enes reli´es `a la r´eaction chimique, tel le front de flamme. Puisque la pr´ediction des polluants associ´es `a ces ´equipements est d´ependante de la qualit´e de la solution num´erique pour plusieurs champs scalaires, la qualit´e de la discr´etisation est de la plus haute importance. La qualit´e du maillage peut ˆetre augment´ee en utilisant de l’adaptation de maillage anisotrope. Cette technique modifie le maillage de mani`ere `a minimiser l’erreur d’interpolation d’une solution ainsi que le nombre de degr´es de libert´e n´ecessaire `a une r´esolution pr´ecise du probl`eme. Les objectifs de cette th`ese par article consistent `a ´elaborer une mesure de la qualit´e des ´el´ements du maillage, qui doit ˆetre coh´erente sur tous les ´el´ements simpliciaux et non simpliciaux. Elle doit quantifier la distorsion anisotrope en taille, ´etirement et orientation par rapport `a une sp´ecification de ces quantit´es. Il est important que cette mesure soit applicable sur des maillages hybrides, puisqu’il est pr´esum´e que les simulations des ´equipements industriels utilisent des mod`eles de turbulence `a loi de parois logarithmiques dans des g´eom´etries complexes, difficiles `a mailler. Dans ces cas, le type de maillage le plus appropri´e est le maillage hybride. La mesure d´evelopp´ee doit ensuite ˆetre utilis´ee dans la construction d’une fonction coˆut et d’un algorithme d’optimisation minimisant cette fonction sur le maillage. Il est d´emontr´e, par des exemples physiques, que l’optimisation de la mesure de qualit´e sur un maillage augmente directement la pr´ecision de la solution num´erique sur celui-ci. Pour terminer, l’algorithme d´evelopp´e est appliqu´e `a un cas de combustion industrielle pour lequel il est d´emontr´e que l’algorithme d’adaptation de maillage propos´e permet d’augmenter la qualit´e de la solution des variables importantes en minimisant les----------ABSTRACT In the field of reacting flow simulations for industrial combustion equipment design, specifically for large utility boilers, computational resources are scarce and the design cycle usually short. The difference between the size of the furnaces to be simulated and the scale of the chemical reactions is of many orders of magnitude, which makes the generation of an adequate mesh to capture features such as the flame front almost impossible. Since the prediction of the pollutants generated by the designed equipment is dependent on the quality of the numerical solution for many scalar fields, the quality of the discretization is of the utmost importance. Mesh quality can be increased by using anisotropic mesh adaptation. This technique modifies the mesh so as to minimize the interpolation error and the degrees of freedom to solve the problem. The objectives of this thesis first consist of developing a measure of mesh element quality which must be coherent for simplicial and non-simplicial elements. The measure must quantify the anisotropic distortion of length, stretching and orientation with respect to a specification of those quantities. It is important that the proposed measure be applicable to hybrid meshes since it is assumed that the combustion simulations use turbulence models with logarithmic wall functions in geometries which are complicated to mesh. In those cases, the most appropriate type of mesh to use is a hybrid mesh. The measure must also be constructed in such a way that it can be used in the construction of a cost function that will be minimized in a mesh optimization algorithm. It will be also shown that the minimization of the cost function based on the proposed quality measure directly improves the solution quality, using multiple examples. Lastly, the developed algorithm is applied to a numerical combustion test case with which it is shown that the mesh adaptation process increases solution accuracy for important variables while minimizing the required computational resources

High-fidelity computational modelling of fluid–structure interaction for moored floating bodies

The development and implementation process of a complete numerical framework for high-fidelity Fluid–Structure Interaction (FSI) simulations of moored floating bodies using Computational Fluid Dynamics (CFD) with the Finite Element Method (FEM) is presented here. For this purpose, the following three main aspects are coupled together: Two-Phase Flow (TPF), Multibody Dynamics (MBD), and mooring dynamics. The fluid–structure problem is two-way and fully partitioned, allowing for high modularity of the coupling and computational efficiency. The Arbitrary Lagrangian–Eulerian (ALE) formulation is used for describing the motion of the mesh-conforming fluid–solid interface, and mesh deformation is achieved with linear elastostatics. Mooring dynamics is performed using gradient deficient Absolute Nodal Coordinate Formulation (ANCF) elements with a two-way mooring–structure coupling and a one-way fluid–mooring coupling. Hydrodynamic loads are applied accurately along mooring cables using the solution of the fluid velocity provided by the TPF solver. For this purpose, fluid mesh elements containing cable nodes that do not conform to the fluid mesh are located with a computationally efficient particle-localisation algorithm. As it is common for partitioned FSI simulations of solids moving within a relatively dense fluid to experience unconditional instability from the added mass effect in CFD, a non-iterative stabilisation scheme is developed here. This is achieved with an accurate and dynamic estimation of the added mass for arbitrarily shaped structures that is then applied as a penalty term to the equations of motion of the solid. It is shown that this stabilisation scheme ensures stability of FSI simulations that are otherwise prone to strong added mass effect without affecting the expected response of structures significantly, even when using fully partitioned fluid–structure coupling schemes. Thorough verification and validation for all aspects of the FSI framework ultimately show that the produced numerical results are in good agreement with experimental data and other inherently stable numerical models, even when complex nonlinear events occur such as vortices forming around sharp corners or extreme wave loads and overtopping on moving structures. It is also shown that the mooring dynamics model can successfully reproduce nonlinearities from high frequency fairlead motions and hydrodynamic loads. The large-scale 3D simulation of a floating semi-submersible structure moored with three catenary lines ties all the models and tools developed here together and shows the capability of the high-fidelity FSI framework to model complex systems robustly and accurately