Reliability Analysis Techniques (Time-Independent)

Abstract

Reliability analysis under uncertainty, which assesses the probability that a system’s performance (e.g., fatigue, corrosion, fracture) meets its marginal value while taking into account various uncertainty sources (e.g., material properties, loads, geometries), has been recognized as having significant importance in product design and process development.

Appendix: A 99-Line MATLAB Code for UDR-Based SRSM

%    A   99-LINE   UDR-BASED   SRSM   CODE   WRITTEN   BY   HU   C.,   Wang   P.,   AND   YOUN   B.D.      % function    UDR_RS() clear    all;    close    all; u   =   [0.4    0.4];                      %% Mean   vector   of   random   variables s   =   [0.01 0.01];                     %% Standard   deviation   vector

%========================  Generate MCS samples  =========================% ns = 5000000;                     %% Number of MCS samples nv = length(u);                   %% Number of input random variables xs = zeros(nv,ns);                %% Initialization of MCS sample vector for k = 1:nv       xs(k,:) = normrnd(u(k),s(k),1,ns); end

%=====================  Obtain Univariate Samples  =======================% [output,input,gg] = UDR_sampling(u,s);

%================  Compute Univariate Response Surface  ==================% % Step 1: obtain univariate component function values uniComp = zeros(nv,ns); for k = 1:nv      uniComp(k,:) = interp1(input(k,:),output(k,:),xs(k,:),′spline′); end % Step 2: combine univariate responses with UDR formula zz = squeeze(uniComp(:,:)); response_URS = sum(zz,1)-(nv-1)*gg;

%===============  Compute True Responses by Direct MCS  ==================% response_true = findresponse(xs);

%==================  Conduct Reliability Analysis  =======================% rel_URS = length(find(response_URS < 0))/ns rel_true = length(find(response_true < 0))/ns

%===============  Plot Probability Density Functions  ====================% % Plot PDF computed from univariate response surface figure(′DefaultAxesFontSize′,16) D = response_URS; [cn,xout] = hist(D,100); sss = sum(cn); unit=(xout(100)-xout(1))/99; for k = 1:100       cn(k)=cn(k)/sss/unit; end plot(xout,cn,′k-′); hold on; clear D cn xout;

% Plot true PDF from direct MCS D = response_true; [cn,xout] = hist(D,100); sss = sum(cn); unit=(xout(100)-xout(1))/99; for k = 1:100     cn(k)=cn(k)/sss/unit; end plot(xout,cn,′r*′); hold on; clear D cn xout; legend(′UDR-SRSM′,′MCS′);  xlabel(′{\itG}({\bfX})′); ylabel(′Probability density′);

%==================  Obtain Univariate Samples  ==========================% function [output,input,gg] = UDR_sampling(u,s) u_loc = [-3.0,-1.5,0,1.5,3.0]; %% Sample locations: [u+/-3.0s, u+/-1.5s, u] nv = length(u);                    %% Dimension of the problem m = length(u_loc);              %% Number of samples along each dimension input = zeros(nv,m); for k = 1:nv      % Identify sample location      input(k,:) = u(k) + u_loc*s(k);      % Get response values      xx = u;      for kk = 1:m            xx(k) = input(k,kk);            if isequal(k,1) && isequal(xx,u)  %% Avoid re-evaluating mean value                  output(k,kk) = findresponse(xx);                 gg = output(k,kk);            elseif ~isequal(k,1) && isequal(xx,u)                 output(k,kk) = gg;            else                 output(k,kk) = findresponse(xx);            end      end end

%=================== Define Performance Function ========================% function response = findresponse(xx) if isvector(xx) == 1      response = 0.75*exp(-0.25*(9*xx(1)-2)^2-0.25*(9*xx(2)-2)^2)....            +0.75*exp(-(9*xx(1)+1)^2/49-(9*xx(2)+1)/10)....            +0.50*exp(-0.25*(9*xx(1)-7)^2-0.25*(9*xx(2)-3)^2)....            -0.20*exp(-(9*xx(1)-4)^2-(9*xx(2)-7)^2) - 0.6; else      response = 0.75*exp(-0.25*(9*xx(1,:)-2).^2-0.25*(9*xx(2,:)-2).^2)....            +0.75*exp(-(9*xx(1,:)+1).^2/49-(9*xx(2,:)+1)/10)....            +0.50*exp(-0.25*(9*xx(1,:)-7).^2-0.25*(9*xx(2,:)-3).^2)....            -0.20*exp(-(9*xx(1,:)-4).^2-(9*xx(2,:)-7).^2) - 0.6; end