An Analysis of Finite Volume, Finite Element, and Finite Difference Methods Using Some Concepts from Algebraic Topology by Claudio MattiussiOverviewFinite Element BasicsFinite Volume BasicsSlide 5Thermostatics EquationsSlide 7Balance EquationsConstitutive EquationsKinematic EquationsChains & CochainsSlide 12Slide 13Cochains & CoboundarySlide 15CoboundaryBalance Equations in FVBalance Equations in FESlide 19Balance Equations FE vs. FVSlide 21Constitutive Equations in FVConstitutive Equations in FEStrategies for Discretizing Constitutive EquationsPowerPoint Presentationp-Cochain ApproximationsExampleExample FESlide 29Example FVSlide 31Slide 32Slide 33FV Cell Optimization OverviewCell Optimization Example #1Slide 36Slide 37Slide 38Slide 39Cell Optimization Example #2Slide 41Slide 42Slide 43Cell Optimization SummaryFinite DifferenceSlide 46SummaryAdditional Reference:An Analysis of Finite Volume, Finite Element, and Finite Difference Methods Using Some Concepts from Algebraic TopologybyClaudio MattiussiIsaac DooleyParallel Programming LabCS DepartmentOverview•Finite Element and Finite Volume Methods from an Algebraic Topology perspective•Thermostatics Example in FE and FV formulations•Optimization of Cells in FVFinite Element Basics•Mesh of nodes and elements•Commonly used for structural simulations•Relate nodal displacements to nodal forces•K = element property matrix (stiffness)•q = vector of unknowns (nodal displacements)•Q = vector of nodal forcing parameters•Boundary values provide Q, Elements provide K •Solve for q€ K[ ]q{ }= Q{ }Finite Volume Basics•Commonly used for fluid flows simulations•Subdivide spatial domain into cells•Maintain an approximation of over each cell•In each timestep we update the approximation of q for each cell using an approximation to the flux through the boundary of the cell•Explicit time stepping may be performed•May be formulated with n-point stencil formulas€ q∫Finite Volume Basics•Upwind methods may use only some subset of possible inflow cells since flow is in a fixed direction.• 5-point stencil:QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.Thermostatics Equations•Unknown Temperature Field T•Given Source Field •Can be discretized by separating into a system of equations € σThermostatics EquationsBalance Equations•Generally relate quantities produced in a volume to the outflow and stored amounts•Transition from smooth to discrete takes place without any approximation error•Terminology:–Local = smooth = differential–Global = discrete•Include conservation laws, equilibrium equations, and static balances.•Require use of metric concepts like length, area, volume, angle, etc.•May include physical constants•Depend upon properties of the medium•Also called material equations or equations of stateConstitutive EquationsKinematic Equations•No approximation involved in discrete renderingChains & Cochains•A chain can represent a collection of cells with integer multiplicity.•Interpretation of a chain is a set of oriented domains with weighting function defined over them.•Recall: Boundary of ChainChains & Cochains•A field is represented as a set of discrete values associated with suitable p-cells. No approximation of the field is required.•For thermostatics we have:•Cochains constitute a representation for fields over discretized domains.Kinematic Equations•Can be represented as a cochain:Cochains & Coboundary•A topological equation asserts the equality of two global quantities, one associated with a geometric object, and the other with its boundary.•So we can define a 3-cochain such that:•This can simply be written:€ δQflow(2)Cochains & CoboundaryCoboundary•Coboundary acts as discrete counterpart to div, curl, grad•The definition of coboundary guarantees the conservation of physical quantities possibly expressed by the topological equations will remain in the discretized equationsBalance Equations in FV•In FV we have an enforcement of 3D heat balance equation such as:•This simply is an application to each 3-cell.•No need to average over cellsBalance Equations in FE•In case of weighted residues, we start with continuous balance equation and enforce the corresponding weighted residual equation for each node of the grid:€ div q = σBalance Equations in FE•We can rewrite (31) using a geometric interpretation as:•The balance equations as written by FE:Balance Equations FE vs. FV•Compare (29) in FV with (34) in FEConstitutive Equations•Connect left and right columns of factorization diagram•A bridge between field variables associated with primary cells, and field variables associated with secondary cells•Discretized using exterior derivativeConstitutive Equations in FVConstitutive Equations in FE•We need to use a cochain approximation to describe this transformation from a discrete representation to a local oneStrategies for Discretizing Constitutive Equations•We can use any method to compute a set of coefficients•No alternative choices exist when choosing how to discretize the topological equations € Λijp-Cochain Approximations•P chain approximation represented in diagram by•We should only use approximations or interpolations of global/discrete values associated with a chain•It is common in electromagnetics but bad to interpolate E and H from local nodal vectors. Instead should interpolate with scalar valued p-cochain approximations since global quantities are scalars.€ ≈( p )Example•2D example with orthogonal rectangular grid•Element Edges coincide with primary 1–cells•The four primary 0–cells are nodes of the elementExample FE•Use 0-cochain approximation and define an interpolation function over a cell, the bilinear polynomial suffices•In a local coordinate system for an element we can find the interpolated temperature to beExample FE•Applying interpolated temperature function to differential kinematic equation and the constitutive equation we get a relation which holds within each element:Example FV•For FV we reconstitute the secondary 2-cochain from field function . To do this we perform an integration of on secondary 1-cells.•Use a secondary grid staggered from the primary one.•We can obtain this relation:€ flow(2)˜ Q € ˜ q € ˜ qExample FV•We next write the balance equations:Example FE•The balance equation for FE