Week 9 - FVM Literature Review

Aim:

To write down the major differences between FDM & FVM. Also, describe the need for interpolation schemes and flux limiters in FVM. You can use the 1D linear heat conduction equation as an example.

Theory:

Introduction:

FVM:

• (FVM) is interpreted as a new kind of Galerkin finite element method (FEM), where the same piecewise linear functions are chosen as in both trial and test spaces, and some specific integration rules are adopted. Error analysis is made for the regular Delaunay triangulation involving obtuse triangles separated, to prove optimal convergence rates of the approximate solutions obtained. The new interpretation makes the FVM analysis much easier because we may bypass verification of the nontrivial Ladyzhenskaya–Babuska–Brezzi (LBB) condition by the Petrov–Galerkin FEM in the existing analysis of FVM. More importantly, the new interpretation and the simple FVM analysis enable us to construct easily the combinations of FVM with other popular numerical methods, such as the finite element method (FEM), the finite differential problems (FDM), the Ritz–Galerkin method (RGM), etc., for solving complicated problems of partial differential equations(PDE). For example, for solving singularity problems, the combination of RGM–FVM is superior to the combination of RGM–FDM in the flexibility of arbitrary solution domains, and also superior to the combination of RGM–FEM in substantial saving of CPU time. Since the conservative law of flux may be maintained exactly in the numerical solutions, and since obtuse triangles may be included in the Delaunay triangulation, the FVM and its combinations become very promising for solving elliptic boundary value problems, in particular, those where singularity solutions exist and those where the obeying of conservative law is crucial. The numerical examples of combinations of RGM–FVM are given for solving Motz's problems, to verify the optimal convergence rates. The techniques and analysis of FVM and its combinations can be extended to the convection-diffusion problems. Most importantly, an important aspect of this paper is the possibility to include obtuse triangles in the error analysis.
• A finite difference method (FDM) discretization is based upon the differential form of the PDE to be solved. Each derivative is replaced with an approximate difference formula (that can generally be derived from a Taylor series expansion). The computational domain is usually divided into hexahedral cells (the grid), and the solution will be obtained at each nodal point. The FDM is easiest to understand when the physical grid is Cartesian, but through the use of curvilinear transforms the method can be extended to domains that are not easily represented by brick-shaped elements. The discretization results in a system of equation of the variable at nodal points, and once a solution is found, then we have a discrete representation of the solution.
  A finite volume method (FVM) discretization is based upon an integral form of the PDE to be solved (e.g. conservation of mass, momentum, or energy). The PDE is written in a form which can be solved for a given finite volume (or cell). The computational domain is discretized into finite volumes and then for every volume the governing equations are solved. The resulting system of equations usually involves fluxes of the conserved variable, and thus the calculation of fluxes is very important in FVM. The basic advantage of this method over FDM is it does not require the use of structured grids, and the effort to convert the given mesh in to structured numerical grid internally is completely avoided. As with FDM, the resulting approximate solution is a discrete, but the variables are typically placed at cell centers rather than at nodal points. This is not always true, as there are also face-centered finite volume methods. In any case, the values of field variables at non-storage locations (e.g. vertices) are obtained using interpolation.
• 
  A finite element method (FEM) discretization is based upon a piecewise representation of the solution in terms of specified basis functions. The computational domain is divided up into smaller domains (finite elements) and the solution in each element is constructed from the basis functions. The actual equations that are solved are typically obtained by restating the conservation equation in weak form: the field variables are written in terms of the basis functions, the equation is multiplied by appropriate test functions, and then integrated over an element. Since the FEM solution is in terms of specific basis functions, a great deal more is known about the solution than for either FDM or FVM. This can be a double-edged sword, as the choice of basis functions is very important and boundary conditions may be more difficult to formulate. Again, a system of equations is obtained (usually for nodal values) that must be solved to obtain a solution.
  Comparison of the three methods is difficult, primarily due to the many variations of all three methods. FVM and FDM provide discrete solutions, while FEM provides a continuous (up to a point) solution. FVM and FDM are generally considered easier to program than FEM, but opinions vary on this point. FVM are generally expected to provide better conservation properties, but opinions vary on this point also. If you are trying to decide which method to use, then the best path is probably found by consulting the literature in the specific problem area.

• Similar to other numerical methods developed for the simulation of fluid flow, the finite volume method transforms the set of partial differential equations into a system of linear algebraic equations. Nevertheless, the discretization procedure used in the finite volume method is distinctive and involves two basic steps. In the first step, the partial differential equations are integrated and transformed into balance equations over an element. This involves changing the surface and volume integrals into discrete algebraic relations over elements and their surfaces using an integration quadrature of a specified order of accuracy. The result is a set of semi-discretized equations. In the second step, interpolation profiles are chosen to approximate the variation of the variables within the element and relate the surface values of the variables to their cell values and thus transform the algebraic relations into algebraic equations. The current chapter details the first discretization step and presents a broad review of numerical issues pertaining to the finite volume method. This provides a solid foundation on which to expand in the coming chapters where the focus will be on the discretization of the various parts of the general conservation equation. In both steps, the selected approximations affect the accuracy and robustness of the resulting numerics. It is therefore important to define some guiding principles for informing the selection process.

• Implicit Versus Explicit Numerical Methods:

• As described, once the number of integration points and the type of linearization are defined, the cell-centered finite volume discretization method results in a set of equations with the values of the dependent variables at the cell centers as unknowns. The way these unknowns are organized and solved classifies the adopted computational approach. Numerical solution schemes are often referred to as being either explicit or implicit. An explicit numerical method is one in which the dependent variables are computed directly via already known values. In this case, any discretization operator can be directly evaluated based on the actual variable values. On the other hand, a numerical method is said to be implicit when the dependent variables are treated as unknowns and assembled to form a coupled set of equations which are then solved via special numerical tools using either a direct or an iterative solution algorithm. The conservation equations dealt with in computational fluid dynamics are nonlinear and the implicit approach is most often preferred over the explicit method for solving them. Once the proper discretization and linearization are performed, the Sf n e dCF f t C F Fig. 15.17 A non-orthogonal element 126 5 The Finite Volume Method's last step is to solve the set or system of algebraic equations to get the solution. Solving the corresponding system with a direct equation solver, as discussed in a previous chapter, is not feasible. The iterative approach is more economical and is the most widely, if not the only one, used. In difference from a direct approach, starting from an initial guess an iterative method progresses toward a solution by using results obtained at the end of an iteration as the initial guess for the following iteration. The sequence of events is said to be converging if the intermediate computed solution is approaching the final solution, otherwise, it is said to be diverging. The process is general and is used whether solving one time step in a transient problem or for a final solution in a steady state problem. In fact, adopting an iterative approach to obtain a steady state solution is computationally cheaper than marching in time until a steady state is reached.

• Now that the basis of the numerics of the FVM and its properties have been presented, it is worth outlining the needed mesh support. In all preceding derivations it has been implicitly assumed that certain geometric and topological information is readily available. Even though the description of the finite volume mesh will be the subject of the next two chapters, based on the covered material so far, it has become possible to draw a simplified list of its characteristics. The application of the FVM as a numerical discretization technique necessitates a mesh support that provides a range of information both geometric and topological, as described next. For an element, the needed information includes: its index, its centroid, a list of bounding faces, and a list of neighboring elements. For a face, information is needed about its index, its centroid, its surface vector, a list of neighboring elements (2 for an interior face and one for a boundary face), in addition to a list of vertices that defines it. Also needed is information about the boundaries of the computational domain, i.e., the boundary faces that define each boundary patch. Another issue to be resolved is the orientation of the normal vector to an element face. In the above derivations it was assumed that the normal to all faces of the element were pointing outward. The normal vectors to faces in a computational domain cannot be all pointing outward. Generally any element will have some of its faces with their normal vectors pointing outward while for the remaining faces they will be pointing inward. This indicates that a proper account should be made of the face sign

• FDM:

• Fused Deposition Modeling (FDM) is an additive manufacturing process in which a machine precisely extrudes melted plastic filament to create a part. Parts are very rigid, especially compared to Selective Laser Sintering (SLS), which makes them a great fit for projects with a rigidity requirement. Given that the filament has a thickness of 0.010” (0.254mm), some fine features may not resolve properly. Knowing this, it is important to follow guidelines set forth in this guide to ensure all features are properly resolved. FDM also requires support material (soluble or break-away) for certain features, so it is important to keep this in consideration when designing parts to allow support structures to be removed during post processing.

• Fused deposition modeling (FDM) represents one of the most common techniques for rapid proto-typing in additive manufacturing (AM). This work applies image based thermography to monitor the FDM process in-situ. The nozzle temperature, print speed and print orientation were adjusted during the fabrication process of each specimen. Experimental and numerical analysis were performed on the fabricated specimens. The combination of the layer wise temperature profile plot and temporal plot provide insights for specimens fabricated in x, y and z-axis orientation. For the x-axis orientation build possessing 35 layers, Specimens B16 and B7 printed with nozzle temperature of 225 ➦C and 235 ➦C respectively, and at printing speed of 60 mm/s and 100 mm/s respectively with the former possessing the highest modulus, yield strength, and ultimate tensile strength. For the y-axis orientation build possessing 59 layers, Specimens B23, B14 and B8 printed with nozzle temperature of 215 ➦C, 225 ➦C and 235 ➦C respectively, and at printing speed of 80 mm/s, 80 mm/s and 60 mm/s respectively with the former possessing the highest modulus and yield strength, while the latter the highest ultimate tensile strength. For the z-axis orientation build possessing 1256 layers, Specimens B6, B24 and B9 printed with nozzle temperature of 235 ➦C, 235 ➦C and 235 ➦C respectively, and at printing speed of 80 mm/s, 80 mm/s and 60 mm/s respectively with the former possessing the highest modulus and ultimate tensile strength, while B24 had the highest yield strength and B9 the lowest modulus, yield strength and ultimate tensile strength. The results show that the prints oriented in the y-axis orientation perform relatively better than prints in the x-axis and z-axis orientation.

• FDM is an additive manufacturing technique devised and patented by Scott Crump. The industrialized form of FDM is the property of STRATASYS, Inc. FDM technology shown in Fig. 1.2 is essentially G-code controlled vertical material extrusion process. FDM produces parts fit for mechanical, chemical and biochemical operating conditions (end use parts). Fig. 1.2 presents the basic components of the FDM process. Fused deposition modeling process commences with the relevant slicer software. The slicer software performs the function of receiving the 3-D CAD information of the part in the form of a stereolithography (STL) file. Example of slicer software include, Repertoire and, Catalyst.