Overview of CFD Solution Methodologies

1. Overview of CFD Solution Methodologies

2. Outline • Ingredients • Overview of Solution Methodologies • Finite Difference • Finite Volume (i.e., Control Volume) • Finite Element • Strengths and Weaknesses

3. domain discretization (grid) solution of algebraic equations equation discretization treatment of convection and source terms Ingredients CFD solution method

4. Survey of Methods • Many CFD techniques exist • The most common are: • Finite Difference • Finite Volume or Control Volume • Finite Element • The focus of this talk is to introduce these three • There are certainly many other approaches, including: • control volume/finite element • spectral • spectral element • boundary element • lattice gas • and more!

5. Finite Difference Method (FDM) • Historically, the oldest of the three • Techniques published as early as 1910 by L. F. Richardson • Seminal paper by Courant, Fredrichson and Lewy (1928) derived stability criteria for explicit time stepping • First ever numerical solution: flow over a circular cylinder by Thom (1933) • Scientific American article by Harlow and Fromm (1965) clearly and publicly the idea of “computer experiments” for the first time — CFD is born!!

6. Finite Volume Method (FVM) • Has its roots in the Finite Difference Method • First well-documented use was by Evans and Harlow (1957) at Los Alamos and Gentry, Martin and Daley (1966) • Was attractive because: • while variables may not be continuously differentiable across shocks and other discontinuities, • mass, momentum and energy would always be conserved • Late 70’s, early 80’s saw development of body-fitted grids • By early 90’s, unstructured grid methods had appeared flow compressible flow over a wedge contours of density

7. Finite Element Method (FEM) • Earliest use was by Courant (1943) for solving St. Venant torsion problem • Clough (1960) gave the method its name • Method was refined greatly in the 60’s and 70’s, mostly for analyzing structural mechanics problem • FEM analysis of fluid flow was developed in the mid- to late 70’s coextrusion metal insert contours of velocity magnitude

8. Finite Difference: Basic Methodology • The domain is discretized into a series of grid points • a “structured” (ijk) mesh is required • The governing equations are discretized (converted to algebraic form) • first and second derivatives are approximated by truncated Taylor series expansions • The resulting set of linear algebraic equations is solved iteratively or simultaneously i j j i

9. Finite Difference: Pro’s and Con’s • Advantages: • simple derivation, implementation • Disadvantages: • relatively simple grids • mass, momentum, energy not conserved on coarse grids

10. Finite Volume: Basic Methodology • Divide the domain into control volumes (c.v.’s) • Integrate the differential equation over the control volume and apply the divergence theorem. • To evaluate derivative terms, values at the control volume faces are needed: have to make an assumption about how the value varies. • Result is a set of linear algebraic equations; one for each c.v. • Solve iteratively or simultaneously. • • • • • • • • • • • • • • • • • • • • •

11. Finite Volume: Pro’s and Con’s • Advantages: • basic FV control volume balance does not limit cell shape • mass, momentum, energy conserved even on coarse grids • efficient, iterative solvers well developed • Disadvantages: • Simplest implementation uses 1-D assumptions during differencing of convection/diffusion terms — leads to false diffusion (multi-dimensional approaches now available)

12. Finite Element: Basic Methodology • Domain is divided into elements. • Most FEM methods use some variant of the Method of Weighted Residuals. • we seek an approximate solution to the governing equations • therefore, we seek to minimize the residual (or error) in some weighted sense over the domain • Choose a shape function which is used to interpolate values between node points. • Multiply the governing equations by a weight function and integrate to obtain the “weak” formulation (contains first derivatives, not second). • Solve algebraic equations iteratively or simultaneously.

13. Finite Element: Pro’s and Con’s • Advantages: • multi-dimensional shape functions give geometric flexibility and limit false diffusion • “natural” imposition of flux boundary conditions • mass is conserved locally and globally, momentum is conserved globally and nearly locally • for same solution accuracy, generally requires fewest grid points • Disadvantages: • traditionally used simultaneously solution of all equations (memory and CPU intensive) — iterative solves now available • perhaps more complicated to develop, debug and maintain

14. Summary • Finite volume method and finite element method are now the most popular and successful methods • Each has its own advantages: • FVM seems to enjoy an advantage in memory use and speed for very large problems, higher speed flows and source term dominated flows (like combustion) • FEM solutions can be very accurate using generally smaller grids • Being a node based scheme with natural imposition of traction boundary conditions, FEM seems better suited for deforming mesh free surface calculations