In mechanical engineering, compliant mechanisms are flexible mechanisms that transfer an input force or displacement to another point through elastic body deformation. These are usually monolithic (single-piece) or jointless structures with certain advantages over the rigid-body, or jointed, mechanisms. In this tutorial, you will learn how to change the top3d program to solve compliant mechanism synthesis.
A compliant mechanism problem involves loading cases: input loading case and dummy loading case. The code also needs to implement a new objective function and its corresponding sensitivity analysis. To demonstrate this implementation, let us consider a three-dimensional force inverter problem as shown in figure below. With an input load defined in the positive direction, the design goal is to maximize the negative horizontal output displacement. Both the top face and the side force are imposed with symmetric constraints; i.e., nodes can only move within the plane.

[expand title="Step.1: Define Loading Conditions" expanded="true" tag="h5"] The new loading conditions as well as input and output points are defined as follows:

[matlab firstline="11"]
% [Input Dummy]
il = [0 nelx]; jl = [nely nely]; kl = [0 0];
loadnid = kl*(nelx+1)(nely+1)+il(nely+1)+(nely+1-jl);
loaddof = 3*loadnid(:) - 2;
din = loaddof(1); dout = loaddof(2);


[matlab firstline="22"]

F = sparse(loaddof,[1 2],[1 -1],ndof,2);


[expand title="Step.2: Define Boundary Conditions" expanded="true" tag="h5"]

The boundary conditions are defined as below:

[matlab firstline="15"]
% Top face
[iif,kf] = meshgrid(0:nelx,0:nelz);
fixednid = kf*(nelx+1)(nely+1)+iif(nely+1)+1;
fixeddof_t = 3fixednid(:)-1;
% Side face
[iif,jf] = meshgrid(0:nelx,0:nely);
fixednid = iif
fixeddof_s = 3fixednid(:);
% Two pins
iif = [0 0]; jf = [0 1]; kf = [nelz nelz];
fixednid = kf
fixeddof_p = [3fixednid(:); 3fixednid(:)-1; 3*fixednid(:)-2];
% Fixed DOFs
fixeddof = union(fixeddof_t,fixeddof_s);
fixeddof = union(fixeddof, fixeddof_p);
fixeddof = sort(fixeddof);


[matlab firstline="23"]
% Displacement vector
U = zeros(ndof,2);
freedofs = setdiff(1:ndof,fixeddof);


[expand title="Step.3: Adding external springs" expanded="true" tag="h5"]

The external springs with stiffness $0.1$ are added at input and output points after line 71.

[matlab firstline="72"]
K(din,din) = K(din,din) + 0.1;
K(dout,dout) = K(dout,dout) + 0.1;</pre>


[expand title="Step.4: Changing objective function and sensitivity analysis" expanded="true" tag="h5"]

The expressions of the objective function and sensitivity are modified in lines 74-76.

[matlab firstline="74"]
U1 = U(:,1); U2 = U(:,2);
ce = reshape(sum((U1(edofMat)*KE).U2(edofMat),2),[nely,nelx,nelz]);
c = U(dout,1);
dc = penal


[expand title="Step.5: Changing OC subroutine" expanded="true" tag="h5"]

The convergence criteria for the bi-sectioning algorithm (lines 82-83) is improved by the following lines:

[matlab firstline="82"]
l1 = 0; l2 = 1e9; move = 0.1;
while (l2-l1)/(l1+l2) > 1e-4 && l2 > 1e-40

To improve the convergence stability, the damping factor of OC-method changes from 0.5 to 0.3 and also takes the positive sensitivities into account, then line 85 is changed to:

[matlab firstline="85"]
xnew = max(0,max(x-move,min(1,min(x+move,x.*(max(1e-10,-dc./dv/lmid)).^0.3))));


[expand title="Step.6: Running the program" expanded="true" tag="h5"]

Compliant mechanisms - Force inverter

The final design shown above is promoted by the line in the $latex textsc{Matlab}$:

[matlab light="true"]