An Analysis of Finite Volume, Finite Element, and Finite Difference Methods Using Some Concepts from Algebraic Topology by Claudio Mattiussi

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An Analysis of Finite Volume, Finite Element, and
Finite Difference Methods Using Some Concepts
from Algebraic TopologybyClaudio Mattiussi

• Isaac Dooley
• Parallel Programming Lab
• CS Department

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Overview

• Finite Element and Finite Volume Methods from an
  Algebraic Topology perspective
• Thermostatics Example in FE and FV formulations
• Optimization of Cells in FV

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Finite 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

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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

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Finite Volume Basics

• Upwind methods may use only some subset of
  possible inflow cells since flow is in a fixed
  direction.
• 5-point stencil

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Thermostatics Equations

• Unknown Temperature Field T
• Given Source Field
• Can be discretized by separating into a system of
  equations

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Thermostatics Equations
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Balance 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.

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Constitutive Equations

• 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
  state

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Kinematic Equations

• No approximation involved in discrete rendering

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Chains 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 Chain

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Chains 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.

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Kinematic Equations

• Can be represented as a cochain

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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

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Cochains Coboundary
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Coboundary

• 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 equations

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Balance 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 cells

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Balance 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

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Balance Equations in FE

• We can rewrite (31) using a geometric
  interpretation as
• The balance equations as written by FE

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Balance Equations FE vs. FV

• Compare (29) in FV with (34) in FE

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Constitutive 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 derivative

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Constitutive Equations in FV
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Constitutive Equations in FE

• We need to use a cochain approximation to
  describe this transformation from a discrete
  representation to a local one

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Strategies 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

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p-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.

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Example

• 2D example with orthogonal rectangular grid
• Element Edges coincide with primary 1cells
• The four primary 0cells are nodes of the element

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Example 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 be

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Example FE

• Applying interpolated temperature function to
  differential kinematic equation and the
  constitutive equation we get a relation which
  holds within each element

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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

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Example FV

• We next write the balance equations

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Example FE

• The balance equation for FE requires integrating
  flow density over elements belonging to the
  support of the weighting functions.
• A secondary 2-cell overlaps with four elements
  which we call d(2)

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Example FE

• The balance equation for FE is therefore
• And similarly to FV we get a result involving
  nine temperatures

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FV Cell Optimization Overview

• Is it necessary to choose cells which coincide
  exactly with a staggered secondary grid?
• We will show that a bilinear approximation
  function on a staggered grid is as good as a
  quadratic one on a generic grid.
• Then we will show that optimal biquadratic
  interpolations on the proper grid perform as well
  as a complete third-order polynomial.
• There exists a common misconception that higher
  order methods do not perform well.

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Cell Optimization Example 1

• Note All quadratic functions through two points
  have the same value of the derivative midway
  between the two points.
• A piecewise linear interpolation on staggered
  mesh gives exact value of the derivative at the
  midpoints or nodes.

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Cell Optimization Example 1

• We will compare bilinear approximations to
  quadratic approximations

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Cell Optimization Example 1

• Next we look for a curve lying in the
  element which satisfies at each point s on the
  curve
• Or satisfying this equation equating the flow
  through the curve

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Cell Optimization Example 1

• We examine the parametric curves
• And then we have an expression for
• Substituting into (72) we get an equality

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Cell Optimization Example 1

• (80) can be satisfied for arbitrary by
• This means that the axes of symmetry of the
  primary 2-cells are optimal loci for the
  evaluation of the normal flow and, therefore, for
  the placement of secondary 1-cells
• Using a bilinear interpolation polynomials, we
  obtain the same degree of accuracy obtainable by
  interpolating with complete second-order
  polynomials

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Cell Optimization Example 2

• Similar to previous example, but with 4 cells per
  element
• Now 9 primary 0-cells per element
• We can use a biquadratic approximation polynomial

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Cell Optimization Example 2

• We will determine optimal placement for cells
  using a biquadratic approximation. Optimality is
  determined by comparing to a higher order
  approximation, just as in previous example.
• Want to find a curve where approximations yield
  same results

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Cell Optimization Example 2

• The previous equality (84) gives rise to
• Which is satisfied by

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Cell Optimization Example 2

• We get then three kinds of optimal secondary
  2-cells
• Large square cells interior to the element
  (9-point formula)
• Medium-sized rectangular cells astride to the
  elements (15-point formula)
• Small square cells centered in the element
  vertices (25-point formula)

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Cell Optimization Summary

• Thus in this case, placing 1-cells on the axes of
  symmetry of the primary 2-cells is not an optimal
  choice. Such a choice is common, and leads people
  to ignorantly discount higher order
  approximations in FV.
• Some are under the wrong impression that FV
  performs well with low-order interpolation, but
  not as well for higher order approximations.
• The paper provides a general form for the
  determination of FV Optimal Cells in section 8.1

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Finite Difference

• A more historical approach, which doesnt yield
  great results.
• FD aims to determine a local discrete
  approximation for the operator constituting the
  field equation
• In this case getting the optimal nine-point FD
  formula, which appears to be the traditional FD
  five-point formula for the laplacian

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Finite Difference

• FD is significantly different from FV
• FV optimizes the approximation of the
  constitutive equation, and thereby flow through
  the boundary of the secondary cells
• FD formula constitutes an optimal approximation
  for the laplacian operator in the central point
  of a patch

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Summary

• FE and FV have many similarities in their
  formulation for the thermostatics problem.
• Various types of equations in physics involve
  approximations or exact discretizations.
• Proper choice of cells in FV is important, and
  often done incorrectly.

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Additional Reference

• E. Tonti On the Formal Structure of Physical
  Theories 1975
• PDF downloadable from internet.
• It contains diagrams of the differential forms as
  well as various equations for almost all physical
  theories