Advanced-numerical-method-1

University of Technical Education HCM City, 2013 Object Physics + Physical Principles + Physical Laws Results + Primary solution + Derivative solution + Simulation Mathematics + ODE and PDE + Initial Condition + Boundary Conditions Methods Feedback + Analytic + Numerical - FDM - FEM - MESHLESS,… CABLE CABLE STRUCTURES STRUCTURES PHYSICAL PHYSICALMODEL MODEL MATH MATH MODEL MODEL Phenomenological Phenomenological observation observation State StateVariables Variables Cause-Effect Cause-Effect Relationships Relationships Mathematical Mathematical Model Model Step1 Step1 Phenomenological Phenomenological observation observation of of the the physical physical system system which which needs needs to to be be modeled modeled Step Step 2 Description Description of of the the state state variables variables and and derivative derivative quantities quantities Step Step 3 Clearly Clearly understanding understanding and and identifications identifications of of the the cause-effect cause-effect relationships, relationships, as as well well as as the the conservation conservation equations equations of of quantities quantities related related to to the the state state of of the the system system (Physical (Physical laws, laws, principle) principle) Step Step 4 Derivation Derivation of of the the mathematical mathematical model model MATHEMATICAL MATHEMATICALMODEL MODELNOTATIONS NOTATIONS ▪▪ Independent Independent variables: variables: Generally Generally time time and and spaces spaces ▪▪ State State variables: variables: The The dependent dependent variables variables ▪▪ Consistency: Consistency: Number Number of of unknown unknown dependent dependent variables variables equal equal to to number number of of independent independent equations equations ▪▪ Closed Closed and and Open Open systems: systems: Do Do not not interact interact with with the the outer outer environment environment and and otherwise otherwise ▪▪ Static Static and and Dynamic Dynamic Models: Models: The The state state variable variable depends depends on on the the time time variable variable Otherwise Otherwise the the mathematical mathematical model model is is static static ▪▪ Finite Finite and and Continuous Continuous models: models: Finite Finite ifif the the state state variable variable does does not not depend depend on on the the space space variables variables (ODE) (ODE) Otherwise Otherwise itit is is continuous continuous (PDE) (PDE) CLASSIFICATION CLASSIFICATION OF OFMODELS MODELSAND AND MATHEMATICAL MATHEMATICALPROBLEMS PROBLEMS WITH WITH ODE ODE ▪▪ ODE ODE MODEL MODEL Input Input Output Output Physical Physical Model Model Dynamical Dynamical response response Initial, Initial, limit limit Conditions Conditions bb Input Input Output Output Mechanical Mechanical Model Model Dynamical Dynamical response response Initial, Initial, limit limit Conditions Conditions bb F0 sin( t ) & cx& kx  F0 sin t mx& F0 & x& 2n x& n x  sin t m ▪▪ LINEAR LINEAR MODEL MODELWITH WITH CONSTANT CONSTANT COEFFICIENTS COEFFICIENTS ▪▪ HOMOGENEOUS HOMOGENEOUS MODEL MODEL ▪▪ NONLINEAR NONLINEAR MODELS MODELS ▪▪ NONLINEARLY NONLINEARLYWEAKLY WEAKLYPERTURBED PERTURBED MODELS MODELS CLASSIFICATION CLASSIFICATION OF OFMODELS MODELSAND AND MATHEMATICAL MATHEMATICALPROBLEMS PROBLEMS WITH WITH PDE PDE • The unknown function depending on at least two variables • Contains some partial derivatives of the unknown function PDE Example Example • Luật phi tuyến c2 x y c1e  ln y x   y c1 xc2  ln y  ln x   y c1 xec2 x  ln(y / x) x   Matlab program clear all clc x=[0.1 0.4 0.5 0.6 0.7 0.9]; y=[0.61 0.92 0.99 1.52 1.67 2.03]; %===================== ======= % Baûng số liệu đo đạc %===================== ======= %xx=[]; %yy=[]; %for i=1:6 xx=log(x); yy=log(y); %end su=0; suu=0; sv=0; suv=0; for i=1:6 su=su+xx(i); suu=suu+(xx(i)^2); sv=sv+yy(i); Nội suy theo luật hàm luỹ thừa suv=suv+xx(i)*yy(i); end d=su^2-6*suu; c2=(su*sv-6*suv)/d b=(su*suv-suu*sv)/d c1=exp(b) y = 1.8311 x0.5227 Luật tổ hợp f ( x) c1 f1 ( x)  c2 f2 ( x)   cn fn ( x) n f ( x )  c i f i ( x ) i 1 Dạng ma trận Cực tiểu hàm thặng dư   2A T y  2A T Ac 0  A T A c A T y  c c   Cho m liệu Chọn luật tổ hợp  0.61     0.92   0.99   y   1.52   1.67     2.03      c  A' * A \  A * y  c1  c    c2  Matlab program clear all clc x=[0.1 0.4 0.5 0.6 0.7 0.9]; y=[0.61 0.92 0.99 1.52 1.67 2.03]; A=zeros(6,2); B=zeros(6,1); for i=1:6 A(i,1)=f1(x(i)); A(i,2)=f2(x(i)); B(i,1)=y(i); end c=(A'*A)\(A'*B) function b=f2(x) b=x; function a=f1(x) a=1/x; y 0.0365  2.2177 x x Matlab program Viscosity of a liquid as a function of temperature >> x = [ 20 30 50 55 ] >> y = [ 0.08 0.015 0.009 0.006 0.0055] >> plot(x,y,’*’) >> plot(x,y,’o’) >> plot(x,y,’.’) > x = linspace(0,2*pi,40) > y = sin(x) > plot(x,y) äi suy theo luật đa thức dựa khai triển Taylor Luật đa thức n y a n x  a n  x n   a1 x  a y1 a n x1n  a n  x1n    a1 x1  a y a n x 2n  a n  x 2n    a1 x  a c x b) Luật tổ hợp tuyến tính  c2 x Bảng liệu đo đạc : y n a n x nn  a n  x nn    a1 x n  a  x1n  n  x2     xn  n a) Luaät Parabol c1 x2  c2 x  c3 x1n  x1 1  a n   y1       n a x2 x 1  n    y                     x nn  x n 1  a   y n   1.0 1.5 1.8 2.0 3.0 3.5 4.5  max 0.09815 0.07579 0.06660 0.04985 0.04662 Matlab program plot(alpha,rho,'o',xfit,yfit1,'r',xfit,yfit2,'c') xlabel('alpha') lear all ylabel('rho') lc title(‘rho=f(alpha)') lpha=[1 1.5 1.8 2.0 3.0 3.5 4.5]'; ho= [0.098158 0.075798 0.066604 0.049851 legend(‘ liệu đo đạc','luật parabol',luật tổ hợp') 046624 0.04189 0.0346]'; % luật parabol qua ñieåm: c1x^2+c2x+c3 grid on =[alpha.^2 alpha ones(size(alpha))]; isp(A'*A) isp(A'*rho) =(A'*A)\(A'*rho) % vẽ đồ thò fit=linspace(min(alpha),max(alpha)); fit1=c(1)*xfit.^2+c(2)*xfit+c(3); % luật c1/x+c2x =[1./alpha alpha]; =(A'*A)\(A'*rho); fit=linspace(min(alpha),max(alpha)); fit2=c(1)./xfit+c(2)*xfit; 3.3 Phương pháp nội suy Lagrange * Nội suy Lagrange 1D, 2D, 3D x  xj N (x j )  � ; j 1 xi  x j (e) i n j �i y  yj N (x j , y j )  � � j 1 xi  x j j 1 yi  y j (e) i n x  xj i j � N i( e ) (x j )   ij  � i �j � j �i n j �i z  zj N (x j , y j , z j )  � � � x  x y  y j 1 i j j 1 i j j 1 zi  z j (e) i n j �i x  xj n j �i y  yj n j �i 3.4 Phương pháp nội suy SPLINES Cubic splines 3.5 Áp dụng Baûng Bảngsố sốliệu liệu đo đạc : đo đạc : x y 2.76 2.51 2.44 2.37 2.34 2.24 2.19 2.10 1.93 Luaät đa thức Luật đa thức Luật luỹ thừa Luật luỹ thừa Luật tổ hợp Luật tổ hợp Luật hàm mũ Luật hàm mũ