If displayflag, figure(10) clf display_3D(xPhys) end ![]() The iteration information is displayed and the structural topology can be also displayed during the iteration process depends on users setting.įunction stop = myOutputFcn(x,optimValues,state,displayflag)įprintf(' It.:%5i Obj.:.4f Vol.:%7.3f ch.:%7.3f\n',eration,optimValues.fval. The myOutputFcn is a function that an optimization function calls at each iteration. function = m圜onstrFcn(x)įor more information, see Nonlinear Constraints. In our problem, we only have one equality constraint, i.e., volume fraction constraintĬlick here to learn more about problem statement and constraints. c is a vector that contains the nonlinear inequalities evaluated at x, ceq is a vector that contains the nonlinear equalities evaluated at x, gradc, the gradient of c(x), and gradceq, the gradient of ceq(x). ![]() m圜onstrFcn accepts a vector x and returns the four vectors c, ceq, gradc, gradceq. The function that computes the nonlinear inequality constraints $c(x) \leq 0$ and the nonlinear equality constraints $ceq(x) = 0$. For an example, see fmincon Interior-Point Algorithm with Analytic Hessian. H = diag(Hessf(:)) lambda.ineqnonlin*Hessc įor details, see Hessian. MyHessianFcn computes the Hessian at a point x with Lagrange multiplier structure lambda: function h = myHessianFcn(x, lambda) Since the constraint is linear constraint, the Hessian of the constraintĭetails about the derivation of Hessian corresponding to the objective function is discussed in the paper. In our problem the Hessian of the objective function can be expressed as: The Hessian of the Lagrangian is given by the equation: See Writing Objective Functions for details. % FILTERING AND MODIFICATION OF SENSITIVITIES % OBJECTIVE FUNCTION AND SENSITIVITY ANALYSISĬe = reshape(sum((U(edofMat)*KE).*U(edofMat),2),) Ĭ = sum(sum(sum((Emin xPhys.^penal*(E0-Emin)).*ce))) ĭc = -penal*(E0-Emin)*xPhys.^(penal-1).*ce U(freedofs,:) = K(freedofs,freedofs)F(freedofs,:) In our problem (minimize compliance) the objective is given byĬlick here to learn more about problem statement, objective function and sensitivity analysis. myObjFcn is a function that accepts a vector x and returns a scalar f and the gradient vector gradf(x), the objective function evaluated at x. 'GradObj','on', 'GradConstr','on', 'Hessian','user-supplied', myOutputFcn(x,optimValues,state,displayflag), Define Objective function OPTIONS = optimset('TolX',tolx, 'MaxIter',maxloop, 'Algorithm','interior-point'. Global ce % Shared between myfun and myHessianFcn Click here for more information out fmincon options.PlotFcns: Matlab provides some built-in plot function, plots the function value.OutputFcn: we defined output function in myOutputFcn, which will display iteration information, structural topology corresponding.We will display iteration information in the OutputFcn. Display: display option is turned off.GradObj, GradConstr, Hessian, HessFcn: With analytic expressions of the gradients of the objective function, constraints and Hessian of the Lagrange function, the optimization algorithm will execute faster than numerical approximation.The hessian in ‘sqp’ or ‘active-set’ are approximated by the means of BFGS (Quasi-Newtown method). The reason why we choose ‘interior-point’ instead of others is because ‘interior-point’ accepted user-supplied Hessian of the Lagrange function while ‘sqp’ and ‘active-set’ do not allow user to provide hessian. ![]()
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