Week 9 - FVM Literature Review

Objective: To study and understand

1. The difference between FDM and FVM.
2. Interpolation schemes in FVM.
3. Flux limiters in FVM.

Description:

There are total 4 Numerical Methods:

1. Finite Element Method (FEM)
2. Boundary Element Method (BEM)
3. Finite Difference Method (FDM)
4. Finite Volume Method (FVM)

Efficient technology for CFD with the finite-difference and finite-volume method has become increasingly popular and have been using from much longer periods.

1) Finite Element Method (FEM):

The finite element method (FEM) is a widely used method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.

The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain.The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function.

Studying or analyzing a phenomenon with FEM is often referred to as finite element analysis (FEA).

BASIC CONCEPT:

The subdivision of a whole domain into simpler parts has several advantages:

• Accurate representation of complex geometry
• Inclusion of dissimilar material properties
• Easy representation of the total solution
• Capture of local effects.

Typical work out of the method involves:

1. dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem
2. systematically recombining all sets of element equations into a global system of equations for the final calculation.

FEM mesh created by an analyst prior to finding a solution to a magnetic problem using FEM software. Colors indicate that the analyst has set material properties for each zone, in this case, a conducting wire coil in orange; a ferromagnetic component (perhaps iron) in light blue; and air in grey. Although the geometry may seem simple, it would be very challenging to calculate the magnetic field for this setup without FEM software, using equations alone.

FEM solution to the problem at left, involving a cylindrically shaped magnetic shield. The ferromagnetic cylindrical part is shielding the area inside the cylinder by diverting the magnetic field created by the coil (rectangular area on the right). The color represents the amplitude of the magnetic flux density, as indicated by the scale in the inset legend, red being high amplitude. The area inside the cylinder is the low amplitude (dark blue, with widely spaced lines of magnetic flux), which suggests that the shield is performing as it was designed to.Boundary Element Method (BEM)

2) Boundary Element Method (BEM):

Boundary integral equations are a classical tool for the analysis of boundary value problems for partial differential equations. The term “ boundary element method” (BEM) denotes any method for the approximate numerical solution of these boundary integral equations.The approximate solution of the boundary value problem obtained by BEM has the distinguishing feature that it is an exact solution of the differential equation in the domain and is parametrized by a finite set of parameters living on the boundary.
Advantages:
The BEM have some advantages over other numerical methods like finite element methods
(FEM) or finite differences:
1. Only the boundary of the domain needs to be discretized. Especially in two dimensions where the boundary is just a curve this allows very simple data input and storage methods.
2. Exterior problems with unbounded domains but bounded boundaries are handled as easily as interior problems.
3. In some applications, the physically relevant data are given not by the solution in the interior of the domain but rather by the boundary values of the solution or its derivatives. These data can be obtained directly from the solution of boundary integral equations, whereas boundary values obtained from FEM solutions are in general not very accurate.
4. The solution in the interior of the domain is approximated with a rather high convergence rate and moreover, the same rate of convergence holds for all derivatives of any 0 order of the solution in the domain. There are difficulties, however, if the solution has to be evaluated close to, but not on the boundary.

3) Finite Difference Method (FDM):

Boundary conditions are needed to truncate the computational domain. They represent communication with the surrounding world, which is the part that we do not want included in our simulation. With the finite-difference method, we may easily run into problems handling boundaries for the purpose of defining the boundary conditions. The finite-difference method is the most direct approach to discretize based upon the differential form of the PDE to be solved. We consider a point in space where we take the continuum representation of the equat ions and replace it with a set of discrete equations,called finite-difference equations.

The finite-difference method is typically defined on a regular/structured grid (i.e. the computational domain is usually divided into hexahedral cells (the grid), and the solution will be obtained at each nodal point.) and this fact can be used for very efficient solution methods. Regular grids are useful for very-large-scale simulations on supercomputers. The method is therefore not usually used for irregular CAD geometries, but more often for rectangular or block­ shaped models.

Examples include Formula 1 car computational-fluid-dynamic (CFD) simulations and space-shuttle CFD simulations.The finite-difference method is more difficult to use for handling material discontinuities. In addition, it does not lend itself for local grid refinement or anything similar to "adaptive mesh refinement:' This may be needed to resolve local rapid variations in solutions such as around a corner of a complex shape, as described earlier.

But, if we can fit the simulation in a rectangular or box-shaped geometry using a regular grid, efficient implementations are much easier than for finite-volume methods. If one can overcome the boundary-condition problem on curved boundaries, the method gives very efficient and high-quality results. The finite-difference method is defined dimension per dimension; this makes it easy to increase the "element order" to get higher-order accuracy.

4) Finite Volume Method (FVM):

The finite-volume method is similar to the finite-differencing method in that the CAD modelis first divided into very small but finite-sized elements of geometrically simple shapes. Apart from this, the finite-volume method is very different from the finite-differencing method, beginning with the concept of elements, which are instead referred to as cells.

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) and can handle discontinuities in solutions. 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 finite-volume method's strength is that it only needs to do flux evaluation for the cell boundaries. This also holds for nonlinear problems, which makes it extra powerfulfor robust handling of (nonlinear) conservation laws appearing in transport problems.

The local accuracy of the finite-volume method, such as close to a corner of interest, can be increased by refining the mesh around that corner. However, the functions that approximate the solution when using the finite-volume method cannot be easily made of higher order. This is a disadvantage of the finite-volume method compared to the finite-difference method.

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 centre's 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.vert ices) are obtained using interpolation.

FVM can garuntee the local and the global conservation of a system.

Interpolation schemes:

The approximations of surface and volume integrals require values of the variable at location other than the computational nodes of the control volume.Values at these locations are obtained using interpolation formulae.

• Linear interpolation
• Upwind interpolation
• Quadratic upwind interpolation
• Hybrid,TVD and ENO interpolation

Linear interpolation:

Approximate values of the variable at the control volume face center by the linear interpolation of the values at the two nearest computational nodes.

`f_e=f_E λ_e+f_p (1-λ_e)`

where, `λ_e=(x_e-x_p)/(x_E-x_p )`

It is simple and most widely used.

The linear interpolation is equivalent to the use of central difference formula of the firs order derivative and hence this scheme is also termed as Central Differencing Scheme (CDS).

Average interface values are given by:

`u_(i-1/2)≈(u_(i-1)+u_i)/2` & `u_(i+1/2)≈(u_i+u_(i+1))/2`

`I_c≈v (u_(i+1)-u_(i-1))/2∆x` which is equivalent to CDS. It is said to be equivalent because both these methods are of order of accuracy 2.

Advantages:

• Approximation of gradients has second order of accuracy on uniform grids
• Its formally of first order on non-uniform grids.

Disadvantages:

• It may produce oscillatory solutions in case of large grid space.

Upwind interpolation:

As the name indicates that we use the node in the upwind or upstream of the given node. It is used in case of convect ion dominated problems.

The upwind interpolation for approximating the value of a variable f at the east face of a control volume is given by:

This interpolation scheme is equivalent to using Forward difference method or backward difference method depending on the flow direction.

If the flow is along -X direction, then the values for this at the interface is given by:

It is said to be equivalent because all these three methods are of order of accuracy 1.

Advantages:

• UDS is the only approximation that satisfies boundedness criterion unconditionally irrespective of velocity and grid spacing i.e. never yield oscillatory solutions.

Disadvantages:

• UDS is numerically diffusive

Numerical diffusion is magnified in multi-dimensional problems if flow is oblique to the grid. Peaks or rapid variations in the variables will be smeared out.

• Only first order accurate. Hence very fine grids are required to obtain accurate solutions.

Quadratic upwind interpolation:

It approximates the values of the variable at the CV face centre by Quadratic Interpolation of the values at three nearest computational nodes (1 Down stream node D & 2 Up stream nodes U ,UU).

Quadratic - As it uses 3 nodes

Upwind - As it uses two upstreams

Convective Kinematics - As it is used for convective domain problems

Advantages:

• It is the better version of UDS and CDS because it is more accurate than UDS and more damping in nature than CDS as it uses 3 nodes to solve.
• It is of 3rd order accurate.

Disadvantages:

• The use of QUICK can lead to subtle problems caused by such unbounded results.
• Implementation of boundary conditions can be problemat ic with such higher-order schemes.

Hybrid,TVD and ENO interpolation:

It is the blend of UDS and CDS.The hybrid difference scheme exploits the favourable properties of the upwind and central differencing schemes. It switches to upwind differencing when central differencing produces inaccurate results at high Pe numbers (Peclet number).

The scheme is fully conservative and since the coefficients are always positive it is unconditionally bounded. It satisfies the transport iveness requirement by usingan upwindformulation for large values of Peclet number.