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Compliant mechanism

Example of the design of a compliant mechanism using topology optimization

It considers the following conditions:

  1. maximum output displacement due to input load

  2. constrained minimum input and output stiffness

In this example of compliant mechanism design, no springs are required on input and output nodes. Also convergence is improved compared to Compliant mechanism (springs).

References:

Koppen, S. (2022). Topology optimization of compliant mechanisms with multiple degrees of freedom. PhD thesis, Delft University of Technology. DOI: http://dx.doi.org/10.4233/uuid:21994a92-e365-4679-b6ac-11a2b70572b7

 20 import numpy as np
 21 import pymoto as pym
 22
 23 # Problem settings
 24 nx, ny = 60, 60  # Domain size
 25 xmin, filter_radius, volfrac = 1e-6, 2, 0.3  # Density settings
 26 nu, E = 0.3, 100  # Material properties
 27
 28 # Values on max compliance on input and output for constraints
 29 max_input_compliance = 10.0
 30 max_output_compliance = 10.0
 31
 32 use_volume_constraint = False
 33
 34 if __name__ == "__main__":
 35     # Set up the domain
 36     domain = pym.VoxelDomain(nx, ny)
 37
 38     # Node and dof groups
 39     nodes_left = domain.get_nodenumber(0, np.arange(ny + 1))
 40     nodes_right = domain.get_nodenumber(nx, np.arange(ny + 1))
 41
 42     dofs_right = domain.get_dofnumber(nodes_right, [0, 1], ndof=2).flatten()
 43     dofs_left_x = 2*nodes_left
 44     dof_input = 2*nodes_left[0] + 1  # Input dof for mechanism
 45     dof_output = 2*nodes_left[-1] + 1  # Output dof for mechanism
 46
 47     fixed_dofs = np.union1d(dofs_left_x, dofs_right)
 48
 49     # Setup rhs for two loadcases
 50     f = np.zeros((domain.nnodes * 2, 2), dtype=float)
 51     f[dof_output, 0] = 1.0
 52     f[dof_input, 1] = 1.0
 53
 54     # Signal with design variables
 55     s_x = pym.Signal('x', state=volfrac * np.ones(domain.nel))
 56
 57     # Setup optimization problem
 58     with pym.Network() as fn:
 59         # Density filtering
 60         s_xfilt = pym.DensityFilter(domain, radius=filter_radius)(s_x)
 61         s_xfilt.tag = 'Filtered density'
 62
 63         # Plot the design
 64         pym.PlotDomain(domain, saveto="out/design")(s_xfilt)
 65
 66         # SIMP penalization
 67         s_xsimp = pym.MathExpression(f"{xmin} + {1 - xmin}*inp0^3")(s_xfilt)
 68
 69         # Assembly of stiffness matrix
 70         s_K = pym.AssembleStiffness(domain, e_modulus=E, poisson_ratio=nu, bc=fixed_dofs)(s_xsimp)
 71
 72         # Solve for the displacements
 73         s_u = pym.LinSolve()(s_K, f)
 74
 75         # Output displacement due to force on input
 76         s_uout = pym.EinSum('i,i->')(s_u[:, 1], f[:, 0])
 77
 78         # Objective is the displacement at the output
 79         s_gobj = pym.Scaling(scaling=10.0)(s_uout)
 80         s_gobj.tag = "Objective"
 81
 82         # Input and output compliances
 83         s_compl = pym.EinSum('ij,ij->j')(s_u, f)
 84
 85         # Compliance constraint on input and output
 86         s_gcompl_out = pym.Scaling(scaling=10.0, maxval=max_output_compliance)(s_compl[0])
 87         s_gcompl_in = pym.Scaling(scaling=10.0, maxval=max_input_compliance)(s_compl[1])
 88         s_gcompl_out.tag = "Output compliance constraint"
 89         s_gcompl_in.tag = "Input compliance constraint"
 90
 91         # List of optimization responses: first the objective and all others the constraints
 92         optimization_responses = [s_gobj, s_gcompl_out, s_gcompl_in]
 93
 94         # Add volume constraint if requested
 95         if use_volume_constraint:
 96             # Volume
 97             s_vol = pym.EinSum('i->')(s_xfilt)
 98             s_vol.tag = "Volume"
 99
100             # Volume constraint
101             s_gvol = pym.Scaling(scaling=10.0, maxval=volfrac * domain.nel)(s_vol)
102             s_gvol.tag = "Volume constraint"
103
104             optimization_responses.append(s_gvol)
105
106     # Plot iteration history
107     pym.PlotIter()(*optimization_responses)
108
109     # Optimization
110     pym.minimize_mma(s_x, optimization_responses, verbosity=2)

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