计算流体力学课件 chap05a

1、Chapter 5,Weighted Residual Methods (WRMs),Weighted Residual Methods,Finite Differences Discrete nodal values Differential formulation for each node Taylor series expansion on a structured grid Truncation errors Accuracy: reduce truncation error,Weighted residuals Continuous shape function Integral

2、formulation for each element Minimize weighted residual for an arbitrary control volume Interpolation errors Accuracy: higher-order interpolation, optimize coefficients for minimum residuals,x,z,y,C.V.,Weighted Residual Methods,Assume the solution can be represented analytically (within the finite v

3、olume) Express the solutions in terms of trial functions Finite-volume Finite element Spectral Boundary element Spectral element Global methods full matrices Local methods banded matrices,Weighted Residual Methods,Start with the integral form of governing equations Assume functional form for trial (

4、interpolation, shape) functions Minimize errors (residuals) with selected weighting functions,Power series Fourier series Lagrange Hermite Chebychev,Weighted Residual Methods,Assume certain profile (trial or shape function) between nodes,Residual Weighted Residual,Weighted Residual Methods,In genera

5、l, we deal with the numerical integration of trial or interpolation functions Trial functions constant, linear, quadratic, sinusoidal, Chebychev polynomial, . Weighting functions subdomain, collocation, least square, Galerkin, .,5.1 General Formulation,Weighted Residual Methods (WRMs) Construct an a

6、pproximate solution Steady problems system of algebraic equations for trial function j (x,y,z) Transient problems system of ODEs in time,Chosen to satisfy I.C./B.C.s if possible,Weighted Residual,Consider one-dimensional diffusion equation In general, R 0 with increasing J (higher-order) Weak form i

7、ntegral form, discontinuity allowed (discontinuous function and/or slope),Exact solution Approximation,Weighted Residual Methods,Weak form integral formulation R 0, but “weighted R” = 0 Choices of shape or interpolation functions? Choices of weighting functions?,Subdomain Method,Equivalent to finite

8、 volume method Dm : numerical element (arbitrary control volume) Dm may be overlapped,Collocation Method,Zero residuals at selected locations (xm, ym, zm) No control on the residuals between nodes,Least Square Method,Minimize the square error,Square error R2 0,Galerkin Method,Weighting function = tr

9、ial (interpolation) function For orthogonal polynomials, the residual R is orthogonal to every member of a complete set!,Numerical Accuracy,How do we determine the most accurate method? How should the error be “weighted”? Zero average error? Least square error? Least rms error? Minimum error within

10、selected domain? Minimum (zero) error at selected points? Minimax minimize the maximum error? Some functions have fairly uniform error distributions comparing to the others,5.1.1 Application to an ODE,Consider a simple ODE (Initial value problem) Use global method with only one element Select a tria

11、l function of the form of Automatically satisfy the auxiliary condition aj = constant, not a function of time,Application to an ODE,Consider a cubic interpolation function with N = 3 QUESTION: Which cubic polynomial gives the best fit to the exact (exponential function) solution? Definition of best

12、fit? Zero average error, least square, least rms, ?,Residual,Substitute the trial function into governing equation For cubic interpolation function N = 3 The residual is a cubic polynomial R 0 Determine the optimal values of aj to minimize the error (under pre-selected weighting functions),Subdomain

13、 Method,Zero average error in each subdomain Note: R(0) = 0.0156 0, R(1) = 0.0155 0,D1,D2,D3,Uniform spacing,x0,x1,x2,x3, A=5/18 8/81 11/324; 3/18 20/81 69/324; 1/18 26/81 163/324 A = 0.2778 0.0988 0.0340 0.1667 0.2469 0.2130 0.0556 0.3210 0.5031 b=1/3; 1/3; 1/3 b = 0.3333 0.3333 0.3333 a=Ab a = 1.0

14、156 0.4219 0.2812,Subdomain Method,Cubic approximation function y(x) The residual R(x) is also a cubic polynomial, x1=0:1/30:1/3; R1=(a(1)-1)+(2*a(2)-a(1)*x1+(3*a(3)-a(2)*x1.2-a(3)*x1.3; x2=1/3:1/30:2/3; R2=(a(1)-1)+(2*a(2)-a(1)*x2+(3*a(3)-a(2)*x2.2-a(3)*x2.3; x3=2/3:1/30:1; R3=(a(1)-1)+(2*a(2)-a(1)

15、*x3+(3*a(3)-a(2)*x3.2-a(3)*x3.3; x=0:0.01:1; zero=0*x; H=plot(x1,R1,x2,R2,r-,x3,R3,g,x,zero,:); set(H,LineWidth,3); xlabel(x); ylabel(R); H2=Title(Subdomain Method); set(H2,Fontsize,16);,Zero average error in each subdomain,Net area under each curve = 0,Subdomain Method,Nonuniform subdomains?,D1,D2,

16、D3,x0,x1,x2,x3,D1,D2,D3,x0,x1,x2,x3,Different coefficients for different choices of subdomains,Grid clustering in high-gradient regions,Least Square Method,Minimum square errors over the entire domain For arbitrary N (symmetric matrix),Least Square Method,For cubic interpolation function (N=3) Nonun

17、iform weighting of residuals over the domain,Least Square Method,Cubic trial function R(0) = 0.0131 0, R(1) = 0.0151 0, A=1/3 1/4 1/5; 1/4 8/15 2/3; 1/5 2/3 33/35 A = 0.3333 0.2500 0.2000 0.2500 0.5333 0.6667 0.2000 0.6667 0.9429 b=1/2; 2/3; 3/4 b = 0.5000 0.6667 0.7500 a=Ab a = 1.0131 0.4255 0.2797

18、 x=0:0.01:1; R=(a(1)-1)+(2*a(2)-a(1)*x+(3*a(3)-a(2)*x.2-a(3)*x.3; R(1) ans = 0.0131 R(101) ans = -0.0151,Least Square Method,R(0) = 0.0131 R(1) = 0.0151, H=plot(x,R,m); set(H,LineWidth,3); zero=0*x; H=plot(x,R,m,x,zero,b:); set(H,LineWidth,3); xlabel(x); ylabel(Residual R); H2=Title(Least Square Met

19、hod); set(H2,Fontsize,16);,Weighted average errors = 0 Minimum sqaure error, x=0:0.01:1; w1=1-x; w2=2*x-x.2; w3=3*x.2-x.3; R=(a(1)-1)+(2*a(2)-a(1)*x+(3*a(3)-a(2)*x.2-a(3)*x.3; R1=w1.*R; R2=w2.*R; R3=w3.*R; zero=0*x; subplot(3,2,1); plot(x,w1,m); ylabel(w_1) subplot(3,2,2); plot(x,R1,m,x,zero,k:); yl

20、abel(w_1R) subplot(3,2,3); plot(x,w2,b); ylabel(w_2) subplot(3,2,4); plot(x,R2,b,x,zero,k:); ylabel(w_2R) subplot(3,2,5); plot(x,w3,r); ylabel(w_3); xlabel(x) subplot(3,2,6); plot(x,R3,r,x,zero,k:); ylabel(w_3R); xlabel(x),MATLAB Script file: Least Square Method,Weighted average error = Net area und

21、er curve = 0,Weighted average error = 0,Weighted average error = 0,Galerkin Method,Weighting function = Trial function,Galerkin Method,For cubic interpolation function (N=3) Small weighting of residuals near x = 0 Largest weight for residuals near x = 1,Galerkin Method,Cubic trial function R(0) = 0.

22、0141 0, R(1) = 0.0141 0,Weighting functions,Weighted residuals,R,xR,x2R,W1 = 1,W2 = x,W3 = x2,Galerkin Method,Galerkin Method,Order of approximation: Linear, Quadratic, and Cubic Trial functions,Galerkin Method,Solution error Equation residual,Galerkin Method,Alternative choice of weighting function

23、s More uniform weighting functions Identical to the least square method,Collocation Method,R = L(T) = 0 at collocation points Identical to Galerkin method if the residuals are evaluated at Gauss-Quadrature points x = 0.1127, 0.5, 0.8873,x0,x1,x2,Collocation Method,Zero residuals at collocation points R(0) = R(0.5) = R(1) = 0,But y yexact at collocation points y(1) = 2.7142857 e,Taylor-series Expansion,Truncated Taylor-series R(0) = 0, R(1) = 1/6 = 0.1667 0 Poor approximation at x = 1 Power series