Weighted Residual Methods Introductory Course on Multiphysics - PowerPoint PPT Presentation

Problem definition Weighted Residual Method ODE example Weighted Residual Methods Introductory Course on Multiphysics Modelling T OMASZ G. Z IELI ´ NSKI bluebox.ippt.pan.pl/˜tzielins/ Institute of Fundamental Technological Research of the Polish Academy of Sciences Warsaw • Poland

Problem definition Weighted Residual Method ODE example Outline Problem definition 1 Boundary-Value Problem Boundary conditions

Problem definition Weighted Residual Method ODE example Outline Problem definition 1 Boundary-Value Problem Boundary conditions 2 Weighted Residual Method General idea Approximation Error functions Minimization of errors System of algebraic equations Categories of WRM

Problem definition Weighted Residual Method ODE example Outline Problem definition 1 Boundary-Value Problem Boundary conditions 2 Weighted Residual Method General idea Approximation Error functions Minimization of errors System of algebraic equations Categories of WRM ODE example 3 A simple BVP approached by WRM Numerical solution Another numerical solution

Problem definition Weighted Residual Method ODE example Outline Problem definition 1 Boundary-Value Problem Boundary conditions 2 Weighted Residual Method General idea Approximation Error functions Minimization of errors System of algebraic equations Categories of WRM ODE example 3 A simple BVP approached by WRM Numerical solution Another numerical solution

Problem definition Weighted Residual Method ODE example Boundary-Value Problem Let B be a domain with the boundary ∂ B , and: L ( . ) be a (second order) differential operator, f = f ( x ) be a known source term in B , n = n ( x ) be the unit vector normal to the boundary ∂ B .

Problem definition Weighted Residual Method ODE example Boundary-Value Problem Let B be a domain with the boundary ∂ B , and: L ( . ) be a (second order) differential operator, f = f ( x ) be a known source term in B , n = n ( x ) be the unit vector normal to the boundary ∂ B . Boundary-Value Problem Find u = u ( x ) = ? satisfying PDE L ( u ) = f in B and subject to (at least one of) the following boundary conditions ∂ u ∂ u α u = ˆ u = ˆ u on ∂ B 1 , ∂ x · n = ˆ γ on ∂ B 2 , ∂ x · n + ˆ β on ∂ B 3 , α ( x ) , and ˆ β = ˆ where ˆ u = ˆ u ( x ) , ˆ γ = ˆ γ ( x ) , ˆ α = ˆ β ( x ) are known fields prescribed on adequate parts of the boundary ∂ B = ∂ B 1 ∪ ∂ B 2 ∪ ∂ B 3 Remarks: the boundary parts are mutually disjoint, for f ≡ 0 the PDE is called homogeneous .

Problem definition Weighted Residual Method ODE example Types of boundary conditions There are three kinds of boundary conditions : 1 the first kind or Dirichlet b.c.: u = ˆ on ∂ B 1 , u

Problem definition Weighted Residual Method ODE example Types of boundary conditions There are three kinds of boundary conditions : 1 the first kind or Dirichlet b.c.: u = ˆ on ∂ B 1 , u 2 the second kind or Neumann b.c.: ∂ u ∂ x · n = ˆ γ on ∂ B 2 ,

Problem definition Weighted Residual Method ODE example Types of boundary conditions There are three kinds of boundary conditions : 1 the first kind or Dirichlet b.c.: u = ˆ on ∂ B 1 , u 2 the second kind or Neumann b.c.: ∂ u ∂ x · n = ˆ γ on ∂ B 2 , 3 the third kind or Robin b.c.: ∂ u α u = ˆ ∂ x · n + ˆ β on ∂ B 3 , also known as the generalized Neumann b.c., it can be presented as ∂ u � � ∂ x · n = ˆ γ + ˆ ˆ α u − u on ∂ B 3 . Indeed, this form is obtained for ˆ β = ˆ γ + ˆ α ˆ u .

Problem definition Weighted Residual Method ODE example Outline Problem definition 1 Boundary-Value Problem Boundary conditions 2 Weighted Residual Method General idea Approximation Error functions Minimization of errors System of algebraic equations Categories of WRM ODE example 3 A simple BVP approached by WRM Numerical solution Another numerical solution

Problem definition Weighted Residual Method ODE example General idea of the method Weighted Residual Method (WRM) assumes that a solution can be approximated analytically or piecewise analytically. In general, a solution to a PDE can be expressed as a linear combination of a base set of functions where the coefficients are determined by a chosen method, and the method attempts to minimize the approximation error .

Problem definition Weighted Residual Method ODE example General idea of the method Weighted Residual Method (WRM) assumes that a solution can be approximated analytically or piecewise analytically. In general, a solution to a PDE can be expressed as a linear combination of a base set of functions where the coefficients are determined by a chosen method, and the method attempts to minimize the approximation error . In fact, WRM represents a particular group of methods where an integral error is minimized in a certain way. Depending on this way the WRM can generate: the finite volume method, finite element methods, spectral methods, finite difference methods.

Problem definition Weighted Residual Method ODE example Approximation Assumption : the exact solution, u , can be approximated by a linear combination of N (linearly-independent) analytical functions, that is, N � u ( x ) ≈ ˜ u ( x ) = U s φ s ( x ) s = 1 Here: ˜ u is an approximated solution, and U s are unknown coefficients, the so-called degrees of freedom , φ s = φ s ( x ) form a base set of selected functions (often called as trial functions or shape functions ). This set of functions generates the space of approximated solutions. s = 1 , . . . N where N is the number of degrees of freedom.

Problem definition Weighted Residual Method ODE example Residua or error functions In general, an approximated solution, ˜ u , does not satisfy exactly the PDE and/or some (or all) boundary conditions. The generated errors can be described by the following error functions : 0 the PDE residuum R 0 (˜ u ) = L (˜ u ) − f ,

Problem definition Weighted Residual Method ODE example Residua or error functions In general, an approximated solution, ˜ u , does not satisfy exactly the PDE and/or some (or all) boundary conditions. The generated errors can be described by the following error functions : 0 the PDE residuum R 0 (˜ u ) = L (˜ u ) − f , 1 the Dirichlet condition residuum R 1 (˜ u ) = ˜ u − ˆ u ,

Problem definition Weighted Residual Method ODE example Residua or error functions In general, an approximated solution, ˜ u , does not satisfy exactly the PDE and/or some (or all) boundary conditions. The generated errors can be described by the following error functions : 0 the PDE residuum R 0 (˜ u ) = L (˜ u ) − f , 1 the Dirichlet condition residuum R 1 (˜ u ) = ˜ u − ˆ u , 2 the Neumann condition residuum u ) = ∂ ˜ u R 2 (˜ ∂ x · n − ˆ γ ,

Problem definition Weighted Residual Method ODE example Residua or error functions In general, an approximated solution, ˜ u , does not satisfy exactly the PDE and/or some (or all) boundary conditions. The generated errors can be described by the following error functions : 0 the PDE residuum R 0 (˜ u ) = L (˜ u ) − f , 1 the Dirichlet condition residuum R 1 (˜ u ) = ˜ u − ˆ u , 2 the Neumann condition residuum u ) = ∂ ˜ u R 2 (˜ ∂ x · n − ˆ γ , 3 the Robin condition residuum u ) = ∂ ˜ u u − ˆ R 3 (˜ ∂ x · n + ˆ α ˜ β .

Problem definition Weighted Residual Method ODE example Minimization of errors Requirement : Minimize the errors in a weighted integral sense � � � � 0 1 2 3 R 0 (˜ u ) ψ r + R 1 (˜ u ) ψ r + R 2 (˜ u ) ψ r + R 3 (˜ u ) ψ r = 0 . B ∂ B 1 ∂ B 2 ∂ B 3 � 0 � 1 � 2 � 3 � � � � Here, , , , and ( r = 1 , . . . M ) are sets of weight ψ r ψ r ψ r ψ r functions . Note that M weight functions yield M conditions (or equations) from which to determine the N coefficients U s . To determine these N coefficients uniquely we need N independent conditions (equations).

Problem definition Weighted Residual Method ODE example Minimization of errors Requirement : Minimize the errors in a weighted integral sense � � � � 0 1 2 3 R 0 (˜ u ) ψ r + R 1 (˜ u ) ψ r + R 2 (˜ u ) ψ r + R 3 (˜ u ) ψ r = 0 . B ∂ B 1 ∂ B 2 ∂ B 3 � 0 � 1 � 2 � 3 � � � � Here, , , , and ( r = 1 , . . . M ) are sets of weight ψ r ψ r ψ r ψ r functions . Note that M weight functions yield M conditions (or equations) from which to determine the N coefficients U s . To determine these N coefficients uniquely we need N independent conditions (equations). Now, using the formulae for residua results in � ∂ ˜ ∂ ˜ � 3 � � � u � u 0 1 2 L (˜ u ) ψ r + ˜ ψ r + ∂ x · n ψ r + ∂ x · n + ˆ α ˜ ψ r u u B ∂ B 1 ∂ B 2 ∂ B 3 � � � � 0 1 2 3 ˆ = f ψ r + u ˆ ψ r + γ ˆ ψ r + β ψ r . B ∂ B 1 ∂ B 2 ∂ B 3