Mesh refinement

In most engineering problems the boundary of the problem domain includes curved sections. The discretization of domains with curved boundaries using meshes that consist of elements with straight sides inevitably involves some error. This type of discretization error can obviously be reduced by mesh refinements. However, in general, it cannot be entirely eliminated unless finite elements which themselves have curved sides are used. 

All numerical computations inevitably involve round-off errors. This error increases as the number of calculations in the solution procedure is increased. Therefore, in practice, successive mesh refinements that increase the number of finite element calculations do not necessarily lead to more accurate solutions. However, one may assume a theoretical situation where the rounding error is eliminated. In this case successive reduction in size of elements in the mesh should improve the accuracy of the finite element solution. Therefore, using a P C" element with sufficient orders of interpolation and continuity, at the limit (i.e. when element dimensions tend to zero), an exact solution should be obtaiiied. This has been shown to be true for linear elliptic problems (Strang and Fix, 1973) where an optimal convergence is achieved if the following conditions are satisfied ... 

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

The comparison between the finite element and analytical solutions for a relatively small value of a - 1 is shown in Figure 2.25. As can be seen the standard Galerkin method has yielded an accurate and stable solution for the differential Equation (2.80). The accuracy of this solution is expected to improve even further with mesh refinement. As Figmre 2.26 shows using a = 10 a stable result can still be obtained, however using the present mesh of 10 elements, for larger values of this coefficient the numerical solution produced by the standard... 

Computational accuracy can be dramatically improved by dynamically adding elements where they minimize the error. For example, more elements ean be added in the neighborhood of a strong gradient in the velocity to help resolve shocks and vortex sheets. Elements may be removed from regions of smooth flow to minimize the computational cost without degrading the overall accuracy. The concept is shown in Fig. 9.6 where a finer mesh overlays the original mesh. This mesh refinement can be carried out to as many levels as necessary [15], [16], [17]. 

M.J. Berger and J. Olinger, Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations, J. Comput. Phys. S3 (1984). 

M.J. Berger and P. Colella, Local Adaptive Mesh Refinement for Shock Hydrodynamics, UCRL-97196, Lawrence Livermore National Laboratory, Livermore, CA, 1987. 

Computational fluid dynamics (CFD) programs are more specialized, and most have been designed to solve sets of equations that are appropriate to specific industries. They can then include approximations and correlations for some features that would be difficult to solve for directly. Four major packages widely used are Fluent (http //www.fluent.com/), CFX (now part of ANSYS), Comsol Multiphysics (formerly FEMLAB) (http //www.comsol.com/), and ANSYS (http //www.ansys.com/). Of these, Comsol Multiphysics is particularly useful because it has a convenient graphical-user interface, permits easy mesh generation and refinement (including adaptive mesh refinement), allows the user to add phenomena and equations easily, permits solution by continuation methods (thus enhancing... 

Three different subcooled impact conditions under which experiments were conducted and reported in the literature are simulated in this study. They are (1) K-heptane droplets (1.5 mm diameter) impacting on the stainless steel surface with We — 43 (Chandra and Avedisian, 1991), (2) 3.8 mm water droplets impacting on the inconel surface at a velocity of 1 m/s (Chen and Hsu, 1995), and (3) 4.0 mm water droplets impacting on the copper surface with We — 25 (Inada et al., 1985). The simulations are conducted on uniform Cartesian meshes (Ax = Ay — Az — A). The mesh size (resolution) is determined by considering the mesh refinement criterion in Section V.A. The mesh sizes in this study are chosen to provide a resolution of CPR =15. 

Step-change in wall boundary condition It was anticipated that the discontinuity in wall temperature at x = 0, and the resulting steep local gradients, would lead to a locally poor approximation which might have adverse effects further downstream. It was soon found that mesh refinement in the axial direction improved the results considerably over the use of an equally-spaced mesh, whereas mesh refinement in the radial direction had little effect, and a fairly coarse uniform radial mesh was always found to be adequate. 

The simulation of non-Newtonian fluid flow is significantly more complex in comparison with the simulation of Newtonian fluid flow due to the possible occurrence of sharp stress gradients which necessitates the use of (local) mesh refinement techniques. Also the coupling between momentum and constitutive equations makes the problem extremely stiff and often time-dependent calculations have to be performed due to memory effects and also due to the possible occurrence of bifurcations. These requirements explain the existence of specialized (often FEM-based) CFD packages for non-Newtonian flow such as POLYFLOW. 

When studying the stability of the steady-state, time-dependent calculations are needed (see [7]). It can also be used as a simple method to compute the steady-state solution. A time-dependent approach using the Lesaint-Ravian technique for the normal stress components and the Baba-Tabata scheme for the shear stress component is developed by Saramito and Piau ([34]). This method allows one to obtain rapidly stationary solutions of the PTT models. Convergence with mesh refinement is obtained as well as oscillation-free solutions. 

First, the elastic stress distributions of the un-notched specimens are obtained from a finite element analysis. For the PI un-notched specimen, the discrepancy between the finite element and the analytical result is very small (about 0.01%), thus validating the finite element calculation in terms of accuracy through the meshing and the type of element used. Therefore a similar calculation is conducted on the G1 un-notched specimen where the span to height ratio is smaller. The mismatch on the maximum stresses at the bottom and at the top of the beam between the finite element calculations and the analytical solution is 0.74% in tension and 0.79% in compression (and remains constant upon further mesh refinement). This estimation of the stress distribution is then used for the following evaluation of the stress intensity factor. 

I.P.Ivrissimtzis, N.A.Dodgson and M.A.Sabin A generative classification of mesh refinement rules with lattice transformations. CAGD... 

Solution adaptive mesh refinement is another powerful feature that enables faster and more-accurate results. This method allows the user to modify the grid interactively based on intermediate solutions so that grid points can be put where they are required and can make the most contribution to the solution accuracy. Although not new, solution adaptive mesh refinement has become increasingly easy to use due to more-sophisticated user interfaces and the ability to handle hybrid meshes in the new generation of CFD packages. 

Berger M, Oliger J (1984) Adaptive mesh refinement for hyperbolic partial differential equations. J ComputPhys 53 484-512... 

Effect of Peclet number and the necessity for mesh refinement Viscosity depending upon concentration or temperature ... 

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