Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part I—a framework for surface mesh optimization

Structured mesh quality optimization methods are extended to optimization of unstructured triangular, quadrilateral, and mixed finite element meshes. New interpretations of well‐known nodally based objective functions are made possible using matrices and matrix norms. The matrix perspective also suggests several new objective functions. Particularly significant is the interpretation of the Oddy metric and the smoothness objective functions in terms of the condition number of the metric tensor and Jacobian matrix, respectively. Objective functions are grouped according to dimensionality to form weighted combinations. A simple unconstrained local optimum is computed using a modified Newton iteration. The optimization approach was implemented in the CUBIT mesh generation code and tested on several problems. Results were compared against several standard element‐based quality measures to demonstrate that good mesh quality can be achieved with nodally based objective functions. Published in 2000 by John Wiley & Sons, Ltd.     

Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part II—A framework for volume mesh optimization and the condition number of the Jacobian matrix

Three‐dimensional unstructured tetrahedral and hexahedral finite element mesh optimization is studied from a theoretical perspective and by computer experiments to determine what objective functions are most effective in attaining valid, high‐quality meshes. The approach uses matrices and matrix norms to extend the work in Part I to build suitable 3D objective functions. Because certain matrix norm identities which hold for 2×2 matrices do not hold for 3×3 matrices, significant differences arise between surface and volume mesh optimization objective functions. It is shown, for example, that the equality in two dimensions of the smoothness and condition number of the Jacobian matrix objective functions does not extend to three dimensions and further, that the equality of the Oddy and condition number of the metric tensor objective functions in two dimensions also fails to extend to three dimensions. Matrix norm identities are used to systematically construct dimensionally homogeneous groups of objective functions. The concept of an ideal minimizing matrix is introduced for both hexahedral and tetrahedral elements. Non‐dimensional objective functions having barriers are emphasized as the most logical choice for mesh optimization. The performance of a number of objective functions in improving mesh quality was assessed on a suite of realistic test problems, focusing particularly on all‐hexahedral ‘whisker‐weaved’ meshes. Performance is investigated on both structured and unstructured meshes and on both hexahedral and tetrahedral meshes. Although several objective functions are competitive, the condition number objective function is particularly attractive. The objective functions are closely related to mesh quality measures. To illustrate, it is shown that the condition number metric can be viewed as a new tetrahedral element quality measure. Published in 2000 by John Wiley & Sons, Ltd.     

A study of incorporating the multigrid method into the three-dimensional finite element discretization: a modular setting and application

Increasing the efficiency of solving linear/linearized matrix equations is a key point to save computer time in numerical simulation, especially for three-dimensional problems. The multigrid method has been determined to be efficient in solving boundary-value problems. However, this method is mostly linked to the finite difference discretization, rather than to the finite element discretization. This is because the grid relationship between fine and coarse grids was not achieved effectively for the latter case. Consequently, not only is the coding complicated but also the performance is not satisfactory when incorporating the multigrid method into the finite element discretization. Here we present an approach to systematically prepare necessary information to relate fine and coarse grids regarding the three-dimensional finite element discretization, such that we can take advantage of using the multigrid method. To achieve a consistent approximation at each grid, we use A ^2h= I ^2h_h A ^h I ^h_2h and b ^2h= I ^2h_h b ^h, starting from the composed matrix equation of the finest grid, to prepare the matrix equations for coarse grids. Such a process is implemented on an element level to reduce the computation to its minimum. To demonstrate the performance, this approach has been used to adapt two existing three-dimensional finite element subsurface flow and transport models, 3DFEMWATER and 3DLEWASTE, to their multigrid version, 3DMGWATER and 3DMGWASTE, respectively. Two example problems, one for each model, are considered for illustration. The computational result shows that the multigrid method can help solve the example problems very efficiently with our presented modular setting. © 1998 John Wiley & Sons, Ltd.     

Simulation of ductile tearing by the BEM with 2D domain discretization and a local damage model

A numerical procedure based on the Boundary Element Method with internal cells and dedicated to the simulation of the ductile tearing of thin metal sheets is presented. Plasticity is handled with an integral formulation based on the initial strain approach involving a discretization of the planar domain. Time integration is performed in an implicit way for the local strain-stress relationships while the global algorithm relies on an explicit formulation. Damage is represented by the scalar parameter of the uncoupled local damage model of Rice and Tracey. Within the scope of our applications, the cracks propagate along paths a priori known. As damage spreads, boundary elements are gradually released. Elastoplastic problems with large yielding zones are solved and compared to reference solutions. At last, the ductile tearing of a specimen is addressed. The calibration of the critical damage parameter leads to numerical results in good agreement with the experimental ones.     

Wettability of carbon surfaces by pure molten alkali chlorides and their penetration into a porous graphite substrate

The wettability of graphite and glassy carbon surfaces by pure molten alkali chlorides (NaCl, KCl, RbCl, CsCl) was measured by the sessile drop method. The contact angle was found to decrease with increase of the cation radius of the chloride. Using our measured and available literature data, a new, semi-empirical model is established to estimate the adhesion energy between the 20 alkali halide molten salts and graphite (or glassy carbon). The adhesion energy is found to increase with square of the radius of the cation, and the inverse of the radius of the anion of the salt. The minimum possible value for the surface energy of graphite (and glassy carbon) was found as 150 ± 30 mJ/m². The critical contact angle of spontaneous penetration (infiltration) of the molten chlorides into a porous graphite substrate was found experimentally below 90°, in the interval between 31° and 58°. This is explained by the inner structure of the porous graphite.