Weighted Residual Methods

Transcription

1 Weighted Residual Methods Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy of Sciences Warsaw Poland

2 Outline 1 Problem definition Boundary-Value Problem Boundary conditions

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

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

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

6 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.

7 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 = û on B 1, u u n = ˆγ on B2, x x n + ˆα u = ˆβ on B 3, where û = û(x), ˆγ = ˆγ(x), ˆα = ˆα(x), and ˆβ = ˆβ(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.

8 Types of boundary conditions There are three kinds of boundary conditions: 1 the first kind or Dirichlet b.c.: u = û on B 1,

9 Types of boundary conditions There are three kinds of boundary conditions: 1 the first kind or Dirichlet b.c.: u = û on B 1, 2 the second kind or Neumann b.c.: u x n = ˆγ on B 2,

10 Types of boundary conditions There are three kinds of boundary conditions: 1 the first kind or Dirichlet b.c.: u = û on B 1, 2 the second kind or Neumann b.c.: 3 the third kind or Robin b.c.: u x n = ˆγ on B 2, u x n + ˆα u = ˆβ on B 3, also known as the generalized Neumann b.c., it can be presented as u x n = ˆγ + ˆα( û u ) on B 3. Indeed, this form is obtained for ˆβ = ˆγ + ˆα û.

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

12 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.

13 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.

14 Approximation Assumption: the exact solution, u, can be approximated by a linear combination of N (linearly-independent) analytical functions, that is, N u(x) ũ(x) = U s φ s (x) s=1 Here: ũ 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.

15 Residua or error functions In general, an approximated solution, ũ, 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 (ũ) = L(ũ) f,

16 Residua or error functions In general, an approximated solution, ũ, 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 (ũ) = L(ũ) f, 1 the Dirichlet condition residuum R 1 (ũ) = ũ û,

17 Residua or error functions In general, an approximated solution, ũ, 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 (ũ) = L(ũ) f, 1 the Dirichlet condition residuum R 1 (ũ) = ũ û, 2 the Neumann condition residuum R 2 (ũ) = ũ x n ˆγ,

18 Residua or error functions In general, an approximated solution, ũ, 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 (ũ) = L(ũ) f, 1 the Dirichlet condition residuum R 1 (ũ) = ũ û, 2 the Neumann condition residuum 3 the Robin condition residuum R 2 (ũ) = ũ x n ˆγ, R 3 (ũ) = ũ x n + ˆαũ ˆβ.

19 Minimization of errors Requirement: Minimize the errors in a weighted integral sense R 0 (ũ) ψ r + R 1 (ũ) ψ r + R 2 (ũ) ψ r + R 3 (ũ) ψ r = 0. B B 1 B 2 Here, { ψ 0 } { 1 } { 2 } { 3 } r, ψ r, ψ r, and ψ r (r = 1,... M) are sets of weight 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). B 3

20 Minimization of errors Requirement: Minimize the errors in a weighted integral sense R 0 (ũ) ψ r + R 1 (ũ) ψ r + R 2 (ũ) ψ r + R 3 (ũ) ψ r = 0. B B 1 B 2 Here, { ψ 0 } { 1 } { 2 } { 3 } r, ψ r, ψ r, and ψ r (r = 1,... M) are sets of weight 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 B L(ũ) 0 ψ r + B 1 1 ũ ψ r + B B 2 ũ x n 2 ψ r + 0 = f ψ r + B 1 B 3 1 û ψ r + B 2 B 3 ( ũ x n + ˆα ũ ) 3 ψ r 2 ˆγ ψ r + B 3 3 ˆβ ψ r.

21 System of algebraic equations By applying the approximation ũ = N U s φ s, and using the properties of linear operators, L(ũ) = N U s L(φ s ), s=1 s=1 ũ N x n = φ s U s x n, the following system of algebraic equations is obtained: N A rs U s = B r s=1 s=1 A rs = B B r = B L(φ s ) ψ 0 r + f 0 ψ r + B 1 B 1 φ s 1 û ψ r + 1 ψ r + B 2 B 2 2 ˆγ ψ r + φ s x n ψ 2 r + B 3 3 ˆβ ψ r. B 3 ( ) φs x n + ˆα φ 3 s ψ r,

22 Categories of WRM generated by weight functions There are four main categories of weight functions which generate the following categories of WRM: Subdomain method. Collocation method. Least squares method. Galerkin method.

23 Categories of WRM generated by weight functions Main categories: Subdomain method. Here the domain is divided in M subdomains B r where { 0 1 x B r, ψ r (x) = 0 outside, such that this method minimizes the residual error in each of the chosen subdomains. Note that the choice of the subdomains is free. In many cases an equal division of the total domain is likely the best choice. However, if higher resolution (and a corresponding smaller error) in a particular area is desired, a non-uniform choice may be more appropriate. Collocation method. Least squares method. Galerkin method.

24 Categories of WRM generated by weight functions Main categories: Subdomain method. Collocation method. In this method the weight functions are chosen to be Dirac delta functions 0 ψ r (x) = δ(x x r ). such that the error is zero at the chosen nodes x r. Least squares method. Galerkin method.

25 Categories of WRM generated by weight functions Main categories: Subdomain method. Collocation method. Least squares method. This method uses derivatives of the residual itself as weight functions in the form 0 ψ r (x) = R 0(ũ(x)). U r The motivation for this choice is to minimize B R2 0 of the computational domain. Note that (if the boundary conditions are satisfied) this choice of the weight function implies ( ) R 2 0 = 0 U r for all values of U r. Galerkin method. B

26 Categories of WRM generated by weight functions Main categories: Subdomain method. Collocation method. Least squares method. Galerkin method. In this method the weight functions are chosen to be identical to the base functions. 0 ψ r (x) = φ r (x). In particular, if the base function set is orthogonal (i.e., B φ r φ s = 0 if r s), this choice of weight functions implies that the residual R 0 is rendered orthogonal with the minimization condition for all base functions. B R 0 0 ψ r = 0

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

28 A simple BVP approached by WRM Boundary Value Problem (for an ODE) Find u = u(x) =? satisfying d 2 u dx du = 0 in B = [a, b], 2 dx subject to boundary conditions on B = B 1 B 2 = {a} {b}: u du x=a = û (Dirichlet), dx = ˆγ (Neumann). x=b

29 A simple BVP approached by WRM Boundary Value Problem (for an ODE) Find u = u(x) =? satisfying d 2 u dx du = 0 in B = [a, b], 2 dx subject to boundary conditions on B = B 1 B 2 = {a} {b}: u du x=a = û (Dirichlet), dx = ˆγ (Neumann). x=b WRM approach: Residua for an approximated solution ũ R 0(ũ) = d2 ũ dx dũ 2 dx, R1(ũ) = ũ x=a û, R 2(ũ) = dũ dx ˆγ. x=b

30 A simple BVP approached by WRM Boundary Value Problem (for an ODE) Find u = u(x) =? satisfying d 2 u dx du = 0 in B = [a, b], 2 dx subject to boundary conditions on B = B 1 B 2 = {a} {b}: u du x=a = û (Dirichlet), dx = ˆγ (Neumann). x=b WRM approach: Residua for an approximated solution ũ R 0(ũ) = d2 ũ dx dũ 2 dx, R1(ũ) = ũ x=a û, R 2(ũ) = dũ dx Minimization of weighted residual error b a ˆγ. x=b ( d2ũ dx dũ ) 0 [ (ũ ) ] [( ) ] 1 dũ ψ 2 r + û ψ r + dx x=a dx ˆγ 2 ψ r = 0. x=b

31 A simple BVP approached by WRM Boundary Value Problem (for an ODE) Find u = u(x) =? satisfying d 2 u dx du = 0 in B = [a, b], 2 dx subject to boundary conditions on B = B 1 B 2 = {a} {b}: u du x=a = û (Dirichlet), dx = ˆγ (Neumann). x=b WRM approach: System of algebraic equations: A rs = b a ( d 2 φ s dx 2 ] B r = [û ψ 1 r x=a dφs dx ) N A rs U s = B r, s=1 0 ψ r + ] + [ˆγ ψ 2 r. x=b [φ s 1 ψ r ] x=a [ dφs + dx ] 2 ψ r x=b,

32 Numerical (and exact) solution Boundary limits and values: a = 0, û = 1, b = 1, ˆγ = 2. u(x) a = b = 1 x

33 Numerical (and exact) solution Boundary limits and values: a = 0, û = 1, b = 1, ˆγ = 2. Shape functions (s = 1, 2): { φs } = { 1, e x }. 2.5 u(x) φ 2 (x) = e x φ 1 (x) = a = b = 1 x

34 Numerical (and exact) solution Boundary limits and values: a = 0, û = 1, b = 1, ˆγ = 2. Shape functions (s = 1, 2): { φs } = { 1, e x }. Weight functions (r = 1, 2): { 0 ψ r } = { 1 ψ r } = { 2 ψ r } = { 1, x }. u(x) φ 2 (x) = e x ψ 1 (x) = ψ 1 1 (x) = ψ 2 1 (x) = 1 φ 1 (x) = ψ 2 (x) = ψ 1 2 (x) = ψ 2 2 (x) = x x a = b = 1

35 Numerical (and exact) solution Boundary limits and values: a = 0, û = 1, b = 1, ˆγ = 2. Shape functions (s = 1, 2): { φs } = { 1, e x }. Weight functions (r = 1, 2): { 0 ψ r } = { 1 ψ r } = { 2 ψ r } = { 1, x }. System of equations: [ ] [ ] [ ] 1 (1 + e) U1 3 =. 0 e 2 Coefficients: Approximated solution: U 2 U 1 = 1 2 e, U2 = 2 e u(x) φ 2 (x) = e x ψ 1 (x) = ψ 1 1 (x) = ψ 2 1 (x) = 1 φ 1 (x) = 1 0 ũ(x) = 2ex +e 2 e (exact solution) ψ 2 (x) = ψ 1 2 (x) = ψ 2 2 (x) = x x a = b = 1 ũ = U 1 + U 2 e x = 2ex + e 2 e.

36 Another numerical solution Boundary limits and values: a = 0, û = 1, b = 1, ˆγ = 2. u(x) 2 u(x) = 2ex +e 2 e (exact solution) a = b = 1 x

37 Another numerical solution Boundary limits and values: a = 0, û = 1, b = 1, ˆγ = 2. Shape and weight functions (s = 1, 2, 3): { φs } = { 0 ψ s } = { 1 ψ s } = { 2 ψ s } = { 1, x, x 2 }. u(x) 2 u(x) = 2ex +e 2 e (exact solution) φ 2 (x) = x φ 1 (x) = 1 φ 3 (x) = x 2 a = b = 1 x

38 Another numerical solution Boundary limits and values: a = 0, û = 1, b = 1, ˆγ = 2. Shape and weight functions (s = 1, 2, 3): { φs } = { 0 ψ s } = { 1 ψ s } = { 2 ψ s } = { 1, x, x 2 }. System of equations: Coefficients: U 1 U 2 U 3 3 = 3. 2 U 1 = , U2 =, U3 = Approximated solution: ũ = U 1 + U 2 x + U 3 x 2 = 3 ( 5 + 4x + 4x 2 ). 17 u(x) φ 2 (x) = x ũ(x) = 3 ( x + 4x 2 ) u(x) = 2ex +e 2 e (exact solution) φ 1 (x) = 1 φ 3 (x) = x 2 a = b = 1 x