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--- optimization [2018/09/18 10:21]
mark.naves created
+++ optimization [2022/01/12 13:41] (current)
jan.dejong
@@ Line 1: / Line 1: @@
 ===== Example: Shape optimization of cross-hinge =====
-This example provides a shape optimization of a single cross flexure by using matlab's <code>fmincon</code> function
+This example provides a shape optimization of a single cross flexure by using matlab's ''fmincon'' function. For this optimization a crosshinge is considered with a mass and inertia connected to node 3. Furthermore, the optimization objective is to maximize the first parasitic frequency of the system (note: rotation of the cross-hinge is prescribed to de desired rotation angle. Therefore, the first parasitic frequency of the system is given by the first eigenfrequency). A rotation of 0.5rad, a maximum stress of 300MPa and and a maximum actuation force of 2.5Nm is considered.
+The cost function of the optimization is defined by the inverse of the first eigenfrequency. Additionally, a limit on the maximum stress and actuation force is implemented by applying a penalty on the value of the cost function. The matlab function which calls spacarlight and computes the cost function is defined by:
+<code matlab run_sim.m>
+function [cost, out] = run_sim(P, Silent)% EXAMPLE SCRIPT FOR RUNNING SPACAR LIGHT
+if nargin == 1
+    Silent = true; %run in silent mode when called by optimizer
+end
+% DESIGN PARAMETERS
+H = P(1);
+B = P(2);
+w = P(3);
+t = P(4);
+% Optimization parameters
+stressmax = 300e6; %Max stress [MPa]
+Mmax = 2.5; %Max actuation moment [Nm]
+Stroke = 0.5; %Max stroke [Rad]
+%% NODE POSITIONS
+%           x y z
+nodes = [   0 0 0;    %node 1
+H 0;    %node 2
+            B H 0;    %node 3
+            B 0 0];   %node 4
+%% ELEMENT CONNECTIVITY
+%               p   q
+elements = [    1   3;  %element 1
+   3;  %element 2
+   4]; %element 3
+%% NODE PROPERTIES
+%node 1
+nprops(1).fix               = true;         %Fix node 1
+%node 3
+nprops(3).rot_z           = Stroke;
+nprops(3).mass            = 1;
+nprops(3).mominertia      = [1e-3 0 0 1e-3 0 1e-3];
+%node 4
+nprops(4).fix               = true;         %Fix node 4
+%% ELEMENT PROPERTIES
+%Property set 1
+eprops(1).elems    = [1 3];            %Add this set of properties to elements 1 and 3
+eprops(1).emod     = 210e9;            %E-modulus [Pa]
+eprops(1).smod     = 70e9;             %G-modulus [Pa]
+eprops(1).dens     = 7800;             %Density [kg/m^3]
+eprops(1).cshape   = 'rect';           %Rectangular cross-section
+eprops(1).dim      = [w t];   %Width: 50 mm, thickness: 0.2 mm
+eprops(1).orien    = [0 0 1];          %Orientation of the cross-section as a vector pointing along "width-direction"
+eprops(1).nbeams   = 2;                %Number of beams used to model this element
+eprops(1).flex     = 1:6;              %Consider all deformation modes (1 to 6) as flexible
+eprops(1).color    = 'grey';           %Color
+eprops(1).opacity  = 0.7;              %Opacity
+eprops(1).cw       = true;             %Constrain warping at nodes (typical condition for flexure clamped at both ends). For a "free" end, warping is typically not constrained
+%Property set 2
+eprops(2).elems    = 2;                %Add this set of properties to element 2
+eprops(2).dens     = 3000;             %Density [kg/m^3]
+eprops(2).cshape   = 'rect';           %Rectangular cross-section
+eprops(2).dim      = [w 10e-3];    %Width: 50 mm, thickness: 10 mm
+eprops(2).orien    = [0 0 1];          %Orientation of the cross-section as a vector pointing along "width-direction"
+eprops(2).nbeams   = 1;                %1 beam for simulating this element (as it is rigid an no more elements are required)
+eprops(2).color    = 'darkblue';
+%% OPTIONAL ARGUMENTS
+opt.filename    = 'crosshinge';     %Filename
+opt.calccompl   = false;            %Disable calculation of compliance matrices (can reduce computation time for large simulations)
+opt.silent      = Silent;            %Run in silent mode
+%% CALL SPACAR_LIGHT AND COMPUTE COST FUNCTION
+try
+    out = spacarlight(nodes, elements, nprops, eprops, opt);
+    cost = 1/min(out.freq(1,:)); %cost function (inverse of first frequency )
+    stress = max(out.stressmax); %max stress
+    Mact = max(out.node(3).Mreac(3,:));%max actuation moment
+    %add penalty if limits are exceeded
+    if stress > stressmax
+        cost = cost * (1 + (stress-stressmax)/stressmax)^5;
+    end
+    if Mact > Mmax
+        cost = cost * (1 + (Mact-Mmax)/Mmax)^5;
+    end
+catch %if sumulation fails, cost is inf
+    cost = inf;
+end
+</code>
+An example to conduct the optimization is provided below, taking into account additional boundary conditions on the design parameters of the cross-hinge. See matlab's documentation on ''fmincon'' for more information.
+<code matlab optimization.m>
+%filename
+simulation = @run_sim;
+%starting point
+x0 = [0.1;  0.1;  50e-3; 0.2e-3];
+%constraints
+b1 = [0.05; 0.05; 25e-3; 0.1e-3];   % Lower bound (b1 < x)
+b2 = [0.2;  0.2;  75e-3; 1e-3  ];       % Upper bound (x < b2)
+A = [-eye(4); eye(4)];
+b = [-b1;b2];
+%optimization
+x_opt = fmincon(simulation, x0, A, b);
+%show results
+[cost, out] = run_sim(x_opt,false);
+</code>
+//Note: prescribing input rotation is only supported for SPACAR light version 1.27 or higher.//

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