Overview of CFD Solution MethodologiesOutlineIngredientsSurvey of MethodsFinite Difference Method (FDM)Finite Volume Method (FVM)Finite Element Method (FEM)Finite Difference: Basic MethodologyFinite Difference: Pro’s and Con’sFinite Volume: Basic MethodologyFinite Volume: Pro’s and Con’sFinite Element: Basic MethodologyFinite Element: Pro’s and Con’sSummary© Fluent Inc. 01/14/19J1Fluids ReviewTRN-1998-004Overview of CFD Solution Methodologies© Fluent Inc. 01/14/19J2Fluids ReviewTRN-1998-004OutlineIngredientsOverview of Solution MethodologiesFinite DifferenceFinite Volume (i.e., Control Volume)Finite Element Strengths and Weaknesses© Fluent Inc. 01/14/19J3Fluids ReviewTRN-1998-004IngredientsCFD solution methodequation discretizationdomain discretization (grid)solution of algebraic equationstreatment of convection and source terms© Fluent Inc. 01/14/19J4Fluids ReviewTRN-1998-004Survey of MethodsMany CFD techniques existThe most common are:Finite DifferenceFinite Volume or Control VolumeFinite ElementThe focus of this talk is to introduce these threeThere are certainly many other approaches, including:control volume/finite elementspectralspectral elementboundary elementlattice gasand more!© Fluent Inc. 01/14/19J5Fluids ReviewTRN-1998-004Finite Difference Method (FDM)Historically, the oldest of the threeTechniques published as early as 1910 by L. F. RichardsonSeminal paper by Courant, Fredrichson and Lewy (1928) derived stability criteria for explicit time steppingFirst 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!!© Fluent Inc. 01/14/19J6Fluids ReviewTRN-1998-004Finite Volume Method (FVM)Has its roots in the Finite Difference MethodFirst 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 conservedLate 70’s, early 80’s saw development of body-fitted gridsBy early 90’s, unstructured grid methods had appearedflowcompressible flow over a wedgecontours of density© Fluent Inc. 01/14/19J7Fluids ReviewTRN-1998-004Finite Element Method (FEM)Earliest use was by Courant (1943) for solving St. Venant torsion problemClough (1960) gave the method its nameMethod was refined greatly in the 60’s and 70’s, mostly for analyzing structural mechanics problemFEM analysis of fluid flow was developed in the mid- to late 70’s coextrusionmetal insertcontours of velocity magnitude© Fluent Inc. 01/14/19J8Fluids ReviewTRN-1998-004Finite Difference: Basic MethodologyThe domain is discretized into a series of grid pointsa “structured” (ijk) mesh is requiredThe governing equations are discretized (converted to algebraic form)first and second derivatives are approximated by truncated Taylor series expansionsThe resulting set of linear algebraic equations is solved iteratively or simultaneouslyijij© Fluent Inc. 01/14/19J9Fluids ReviewTRN-1998-004Finite Difference: Pro’s and Con’sAdvantages:simple derivation, implementationDisadvantages:relatively simple gridsmass, momentum, energy not conserved on coarse grids© Fluent Inc. 01/14/19J10Fluids ReviewTRN-1998-004Finite Volume: Basic MethodologyDivide 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.•••••••••••••••••••••© Fluent Inc. 01/14/19J11Fluids ReviewTRN-1998-004Finite Volume: Pro’s and Con’sAdvantages:basic FV control volume balance does not limit cell shapemass, momentum, energy conserved even on coarse gridsefficient, iterative solvers well developedDisadvantages:Simplest implementation uses 1-D assumptions during differencing of convection/diffusion terms — leads to false diffusion (multi-dimensional approaches now available)© Fluent Inc. 01/14/19J12Fluids ReviewTRN-1998-004Finite Element: Basic MethodologyDomain is divided into elements.Most FEM methods use some variant of the Method of Weighted Residuals.we seek an approximate solution to the governing equationstherefore, we seek to minimize the residual (or error) in some weighted sense over the domainChoose 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.© Fluent Inc. 01/14/19J13Fluids ReviewTRN-1998-004Finite Element: Pro’s and Con’sAdvantages:multi-dimensional shape functions give geometric flexibility and limit false diffusion“natural” imposition of flux boundary conditionsmass is conserved locally and globally, momentum is conserved globally and nearly locallyfor same solution accuracy, generally requires fewest grid pointsDisadvantages:traditionally used simultaneously solution of all equations (memory and CPU intensive) — iterative solves now availableperhaps more complicated to develop, debug and maintain© Fluent Inc. 01/14/19J14Fluids ReviewTRN-1998-004SummaryFinite volume method and finite element method are now the most popular and successful methodsEach 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 gridsBeing a node based scheme with natural imposition of traction boundary conditions, FEM seems better suited for deforming mesh free surface
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