Finite element methods meshes

Many multiscale methods have been developed across different disciplines. Consequently, much needs to be done in the fundamental theory of multiscale numerical methods that applies across these disciplines. One method is famous in structural materials problems the quasi-continuum method of Tadmor, Ortiz, and Philips. It links the atomistic and continuum models through the finite element method by doing a separate atomist structural relaxation calculation on each cell of the finite element method mesh, rather than using empirical constitutive information. Thus, it directly and dynamically incorporates atomistic-scale information into the deterministic scale finite element method. It has been nsed mainly to predict observed mechanical properties of materials on the basis of their constituent defects. 

Mesh refinement - h- and p-versions of the finite element method... 

The trial functions in the finite element method are not limited to linear ones. Quadratic functions and even higher-order functions are frequently used. The same considerations hold as for boundary value problems The higher-order trial functions converge faster, but require more work. It is possible to refine both the mesh h and the power of polynomial in the trial function p in an hp method. Some problems have constraints on some of the variables. For flow problems, the pressure must usually be approximated by using a trial function that is one order lower than the polynomial used to approximate the velocity. 

In the finite element method, Petrov-Galerkin methods are used to minimize the unphysical oscillations. The Petrov-Galerkin method essentially adds a small amount of diffusion in the flow direction to smooth the unphysical oscillations. The amount of diffusion is usually proportional to Ax so that it becomes negligible as the mesh size is reduced. The value of the Petrov-Galerkin method lies in being able to obtain a smooth solution when the mesh size is large, so that the computation is feasible. This is not so crucial in one-dimensional problems, but it is essential in two- and three-dimensional problems and purely hyperbolic problems. 

Stress calculations are carried out by the finite element method. Here, the commercial finite method code ABAQUS (Hibbit, Karlsson, and Sorensen, Inc.) is used. Other codes such as MARC, ANSYS are also available. To calculate the stresses precisely, appropriate meshes and elements have to be used. 2D and shell meshes are not enough to figure out stress states of SOFC cells precisely, and thus 3D meshes is suitable for the stress calculation. Since the division of a model into individual tetrahedral sometimes faces difficulties of visualization and could easily lead to errors in numbering, eight-comered brick elements are convenient for the use. The element type used for the stress simulation here is three-dimensional solid elements of an 8-node linear brick. In the coupled calculation between the thermo-fluid calculation and the stress calculation a same mesh model have to be used. Consequently same discrete 3D meshes used for the thermo-fluid analysis are employed for the stress calculation. Using ABAQUS, the deformations and stresses in a material under a load are calculated. Besides this treatment, the initial and final conditions of models can be set as the boundary conditions and the structural change can thus be treated. 

One major advantage of finite element methods over finite difference methods is the way they naturally incorporate non-uniform meshes. They can therefore be applied to problems with a complex geometry (Stevens et al, 1997), for example elevated and recessed electrodes (Ferrigno et ah, 1997), and, in principle, simulation of rough electrodes. On the downside, finite element methods are more complex to program, especially when simulating chemical steps, and result in a linear system of equations which is not neatly banded. 

An analogous result is valid for continuous approximations of r when up winding is performed by the streamline upwinding Petrov-Galerkin method (SUPG) [104]. The same is true for finite element methods based on a quadrangular mesh [105]. 

A commercial Computational Fluid Dynamics package (FIDAP Version 7.6, Fluid Dynamics International, Evanston, IL) based on the finite element method was used to solve the governing continuity, momentum and heat transport equations. A mesh was defined with more nodes near the wall and the entrance of the tubular heat exchanger to resolve the larger variations of temperature and velocities near the wall and the entrance. 

Neither the finite element method nor a discretised integration will "catch" eddy or other motion below a certain scale determined by the choice of mesh. Small-scale motion in many cases may be better described as random, in which case the transport of the quantity A is called diffusion. Diffusion can be described by Pick s law, assuming that the flux density /, i.e., the number of "particles" (here a small parcel of a gas), passing a unit square in a given direction, is proportional to the negative gradient of particle concentration n (Bockris and Despic, 2004),... 

Peskin [49] used the Galerkin finite-element method to compute current distribution and shape change for electrodeposition into rectangular cavities. A concentration-dependent overpotential expression including both forward and reserve rate terms was used, and a stagnant diffusion layer was assumed. An adaptive finite-element meshing scheme was used to redefine the problem geometry after each time step. 

Here it suffices to say that the coupling interface between the finite element mesh and the atomic coordinates in the molecular dynamics is accomplished through resolving the near part of the finite element mesh on the atomic coordinates. On either side of the interface, the atoms and the finite element mesh overlap. Finite element cells that intersect the interface and atoms that interact across the interface each contribute to the Hamiltonian at half strength. See Rudd and Broughton (1998) for further discussion of the relationship between molecular dynamics and finite element methods. 

Macroscopic phenomena are described by systems of integro-partial differential algebraic equations (IPDAEs) that are simulated by continuum methods such as finite difference, finite volume and finite element methods ([65] and references dted therein [66, 67]). The commonality of these methods is their use of a mesh or grid over the spatial dimensions [68-71]. Such methods form the basis of many common software packages such as Fluent for simulating fluid dynamics and ABAQUS for simulating solid mechanics problems. 

The finite element method was used for the discretization of the flow equations. Considering the complex kinematics in the coaxial mixer and the associated change of topology at each time step, a new mesh should a priori be built for every topology considered in the time discretization. As a large number of time steps would be required to depict the agitator kinematics accurately, this approach would be a tremendous chore. To alleviate this difficulty, several alternatives have been proposed in the literature ... 

The boundary conditions are applied in the finite element method in a different way than in the finite difference method, and then the linear algebra problem is solved to give the approximation of the solution. The solution is known at the grid points, which are the points between elements, and a form of the solution is known in between, either linear or quadratic in position as described here. (FEMLAB has available even higher order approximations.) The result is still an approximation to the solution of the differential equation, and the mesh must be refined and the procedure repeated until no further changes are noted in the approximation. 

---