Maximum Local Compliance

This loal compliance maximization designs structures with the maximum displacement in one node. This can be used to design MEMS actuators as is discussed at Maximum Local Compliance. An example as how to use the optimization is shown in an example optimization example.py

Density Constraints

Constraints class used to specify the density constraints of the topology optimisation problem. It contains functions for minimum and maximum element density in the upcoming iteration and the magnitude of the volume constraint function itself of the current design. This version of the code is used for the compliant design, local displacement maximisation.

Bram Lagerweij Aerospace Structures and Materials Department TU Delft 2018

class src_Actuator.constraints.DensityConstraint(nelx, nely, move, volume_frac, density_min=0.0, density_max=1.0)

This object relates to the constraints used in this optimization. It can be used for the MMA update scheme to derive what the limit is for all element densities at every iteration. The class itself is not changed by the iterations.

Parameters

• nelx (int) – Number of elements in x direction.

• nely (int) – Number of elements in y direction.

• move (float) – Maximum change in density of an element over 1 itteration.

• volume_frac (float) – Maximum volume that can be filled with material.

• volume_derivative (2D array size(1, nelx*nely)) – Sensitivity of the density constraint to the density in each element.

• density_min (float (optional)) – Minimum density, set at 0.0 if not specified.

• density_max (float (optional)) – Maximum density, set at 0.0 if not specified.

nelx

Number of elements in x direction.

Type

int

nely

Number of elements in y direction.

Type

int

move

Maximum change in density of an element over 1 iteration.

Type

float

volume_frac

Maximum volume that can be filled with material.

Type

float

volume_derivative

Sensitivity of the density constraint to the density in each element.

Type

2D array size(1, nelx*nely)

density_min

Minimum density, set at 0.0 if not specified.

Type

float, optional

density_max

Maximum density, set at 0.0 if not specified.

Type

float, optional

current_volconstrain(x)

Calculates the current magnitude of the volume constraint function:

\[V_{\text{constraint}} = \frac{\sum v_e X_e}{ V_{\max}}-1\]

Parameters

x (2D array size(nely, nelx)) – Density distribution of this iteration.

Returns

curvol – Current value of the density constraint function.

Return type

float

xmax(x)

This function calculates the maximum density value of all elements of this iteration.

Parameters

x (2D array size(nely, nelx)) – Density distribution of this iteration.

Returns

xmax – Maximum density values of this itteration after updating.

Return type

2D array size(nely, nelx)

xmin(x)

This function calculates the minimum density value of all elements of this iteration.

Parameters

x (2D array size(nely, nelx)) – Density distribution of this iteration.

Returns

xmin – Minimum density values of this iteration for the update scheme.

Return type

2D array size(nely, nelx)

Load Cases

This file contains the Load class that allows the generation of an object that contains geometric, mesh, loads and boundary conditions that belong to the load case. This version of the code is meant for local compliant maximization.

Bram Lagerweij Aerospace Structures and Materials Department TU Delft 2018

Parent Load Case

class src_Actuator.loads.Load(nelx, nely, young, Emin, poisson, ext_stiff)

Load parent class that contains the basic functions used in all load cases. This class and its children do cantain information about the load case conciderd in the optimisation. The load case consists of the mesh, the loads, and the boundaries conditions. The class is constructed such that new load cases can be generated simply by adding a child and changing the function related to the geometry, loads and boundaries.

Parameters

• nelx (int) – Number of elements in x direction.

• nely (int) – Number of elements in y direction.

• young (float) – Youngs modulus of the materias.

• Emin (float) – Artifical Youngs modulus of the material to ensure a stable FEA. It is used in the SIMP based material model.

• poisson (float) – Poisson ration of the material.

nelx

Number of elements in x direction.

Type

int

nely

Number of elements in y direction.

Type

int

young

Youngs modulus of the materias.

Type

float

Emin

Artifical Youngs modulus of the material to ensure a stable FEA. It is used in the SIMP based material model.

Type

float

poisson

Poisson ration of the material.

Type

float

dim

Amount of dimensions conciderd in the problem, set at 2.

Type

int

ext_stiff

Extra stiffness to be added to global stiffness matrix. Due to interactions with meganisms outside design domain.

Type

float

alldofs()

Returns a list with all degrees of freedom.

Returns

all – List with numbers from 0 to the maximum degree of freedom number.

Return type

1-D list

displaceloc()

Returns a zero vector, there is supposed to be an 1 implemented at the index where displacment output should be maximised, such that u·l = u_out

Returns

l – Empty for the governing class.

Return type

1-D column array length covering all degrees of freedom

edof()

Generates an array with the position of the nodes of each element in the global stiffness matrix.

Returns

• edof (2-D array size(nelx*nely, 8)) – The list with all elements and their degree of freedom numbers.

• x_list (1-D array len(nelx*nely*8*8)) – The list with the x indices of all ellements to be inserted into the global stiffniss matrix.

• y_list (1-D array len(nelx*nely*8*8)) – The list with the y indices of all ellements to be inserted into the global stiffniss matrix.

fixdofs()

Returns a list with indices that are fixed by the boundary conditions.

Returns

fix – List with all the numbers of fixed degrees of freedom. This list is empty in this parrent class.

Return type

1-D list

force()

Returns an 1D array, the force vector of the loading condition.

Returns

f – Empy force vector.

Return type

1-D column array length covering all degrees of freedom

freedofs()

Returns a list of arr indices that are not fixed

Returns

free – List containing all elemens of alldogs except those that appear in the freedofs list.

Return type

1-D list

lk(E, nu)

Calculates the local siffness matrix depending on E and nu.

Parameters

• E (float) – Youngs modulus of the material.

• nu (float) – Poisson ratio of the material.

Returns

ke – Local stiffness matrix.

Return type

2-D array size(8, 8)

node(elx, ely)

Calculates the topleft node number of the requested element.

Parameters

• elx (int) – X position of the conciderd element.

• ely (int) – Y position of the conciderd element.

Returns

topleft – The node number of the top left node.

Return type

int

nodes(elx, ely)

Calculates all node numbers of the requested element

Parameters

• elx (int) – X position of the conciderd element.

• ely (int) – Y position of the conciderd element.

Returns

• n1 (int) – The node number of the top left node.

• n2 (int) – The node number of the top right node.

• n3 (int) – The node number of the bottom right node.

• n4 (int) – The node number of the bottom left node.

passive()

Retuns three lists containing the location and magnitude of fixed density values

Returns

• elx (1-D list) – X coordinates of all passive elements, empty for the parrent class.

• ely (1-D list) – Y ccordinates of all passive elements, empty for the parrent class.

• values (1-D list) – Density values of all passive elements, empty for the parrent class.

• fix_ele (1-D list) – List with all element numbers that are allowed to change.

Child Load Cases

class src_Actuator.loads.Inverter(nelx, nely, young, Emin, poisson, ext_stiff)

Bases: src_Actuator.loads.Load

This child of the Load class represents a top half of the symetric inverter design used for MEMS actuators. It contains an positive horizontal force at the bottom left corner which causes a negative displacement at the bottom right corner.

No methods are added compared to the parrent class. Only the force, displaceloc and fixdof equations are changed to contain the propper values for the boundary conditions and optimisation objective.

displaceloc()

The maximisation should occur in negative x direction at the bottom right corner. Positive movement is thus in negative x direction.

Returns

l – Value of -1 at the index related to the bottom right node.

Return type

1-D column array length covering all degrees of freedom

fixdofs()

The boundary conditions of this problem fixes the bottom of the desing space in y direction (due to symetry). While the topleft element is fixed in both x and y direction on the left side.

Returns

fix – List with all the numbers of fixed degrees of freedom.

Return type

1-D list

force()

The force vector containts a load in positive x direction at the bottom left corner node.

Returns

f – Value of 1 at the index related to the bottom left node.

Return type

1-D column array length covering all degrees of freedom

Finite Element Solvers

Finite element solvers for the displacement from stiffness matrix, force and adjoin vector. This version of the code is meant for local compliant maximization.

Bram Lagerweij Aerospace Structures and Materials Department TU Delft 2018

Parent Solver

class src_Actuator.fesolvers.FESolver(verbose=False)

This parent FEA class can only assemble the global stiffness matrix and exclude all fixed degrees of freedom from it. This stiffness csc-sparse stiffness matrix is assembled in the gk_freedof method. This class solves the FE problem with a sparse LU-solver based upon umfpack. This solver is slow and inefficient. It is however more robust.

For this local compliance (actuator) maximization this solver solves two problems, the equilibrium and the adjoint problem which will be required to compute the gradients.

Parameters

verbose (bool, optional) – False if the FEA should not print updates

verbose

False if the FEA should not print updates.

Type

bool

displace(load, x, ke, kmin, penal)

FE solver based upon the sparse SciPy solver that uses umfpack.

Parameters

• load (object, child of the Loads class) – The loadcase(s) considered for this optimisation problem.

• x (2-D array size(nely, nelx)) – Current density distribution.

• ke (2-D array size(8, 8)) – Local fully dense stiffness matrix.

• kmin (2-D array size(8, 8)) – Local stiffness matrix for an empty element.

• penal (float) – Material model penalisation (SIMP).

Returns

• u (1-D column array shape(max(edof), 1)) – The displacement vector.

• lamba (1-D column array shape(max(edof), 1)) – Adjoint equation solution.

gk_freedofs(load, x, ke, kmin, penal)

Generates the global stiffness matrix with deleted fixed degrees of freedom. It generates a list with stiffness values and their x and y indices in the global stiffness matrix. Some combination of x and y appear multiple times as the degree of freedom might appear in multiple elements of the FEA. The SciPy coo_matrix function adds them up at the background. At the location of the force introduction and displacement output an external stiffness is added due to stability reasons.

Parameters

• load (object, child of the Loads class) – The loadcase(s) considered for this optimisation problem.

• x (2-D array size(nely, nelx)) – Current density distribution.

• ke (2-D array size(8, 8)) – Local fully dense stiffness matrix.

• kmin (2-D array size(8, 8)) – Local stiffness matrix for an empty element.

• penal (float) – Material model penalisation (SIMP).

Returns

k – Global stiffness matrix without fixed degrees of freedom.

Return type

2-D sparse csc matrix

Child Solvers

class src_Actuator.fesolvers.CvxFEA(verbose=False)

Bases: src_Actuator.fesolvers.FESolver

This parent FEA class can assemble the global stiffness matrix and solve the FE problem with a Supernodal Sparse Cholesky Factorization. It solves for both the equilibrium and adjoin problems.

verbose

False if the FEA should not print updates.

Type

bool

displace(load, x, ke, kmin, penal)

FE solver based upon a Supernodal Sparse Cholesky Factorization. It requires the installation of the cvx module. It solves both the FEA equilibrium and adjoint problems. 1

Parameters

• load (object, child of the Loads class) – The loadcase(s) considerd for this optimisation problem.

• x (2-D array size(nely, nelx)) – Current density distribution.

• ke (2-D array size(8, 8)) – Local fully dense stiffness matrix.

• kmin (2-D array size(8, 8)) – Local stiffness matrix for an empty element.

• penal (float) – Material model penalisation (SIMP).

Returns

• u (1-D column array shape(max(edof), 1)) – The displacement vector.

• lamba (1-D column array shape(max(edof), 1)) – Adjoint equation solution.

References

1

Y. Chen, T. A. Davis, W. W. Hager, S. Rajamanickam, “Algorithm 887: CHOLMOD, Supernodal Sparse Cholesky Factorization and Update/Downdate”, ACM Transactions on Mathematical Software, 35(3), 22:1-22:14, 2008.

class src_Actuator.fesolvers.CGFEA(verbose=False)

Bases: src_Actuator.fesolvers.FESolver

This parent FEA class can assemble the global stiffness matrix and solve the FE problem with a sparse solver based upon a preconditioned conjugate gradient solver. The preconditioning is based upon the inverse of the diagonal of the stiffness matrix.

Recommendations

• Make the tolerance change over the iterations, low accuracy is required for first iteration, more accuracy for the later ones.

• Add more advanced preconditioned.

• Add gpu acceleration.

verbose

False if the FEA should not print updates.

Type

bool

ufree_old

Displacement field of previous iteration.

Type

array len(freedofs)

lambafree_old

Ajoint equation result of previous iteration.

Type

array len(freedofs)

displace(load, x, ke, kmin, penal)

FE solver based upon the sparse SciPy solver that uses a preconditioned conjugate gradient solver, preconditioning is based upon the inverse of the diagonal of the stiffness matrix. Currently the relative tolerance is hardcoded as 1e-5. It solves both the equilibrium and adjoint problems.

Parameters

• load (object, child of the Loads class) – The loadcase(s) considered for this optimisation problem.

• x (2-D array size(nely, nelx)) – Current density distribution.

• ke (2-D array size(8, 8)) – Local fully dense stiffness matrix.

• kmin (2-D array size(8, 8)) – Local stiffness matrix for an empty element.

• penal (float) – Material model penalisation (SIMP).

Returns

• u (1-D array len(max(edof)+1)) – Displacement of all degrees of freedom

• lamba (1-D column array shape(max(edof), 1)) – Adjoint equation solution.

Optimization Module

Topology Optimization class that handles the iterations, objective functions, filters and update scheme. It requires to call upon a constraint, load case and FE solver classes. This version of the code is meant for local compliant maximization (Actuator design).

Bram Lagerweij Aerospace Structures and Materials Department TU Delft 2018

class src_Actuator.topopt.Topopt(constraint, load, fesolver, verbose=False)

This is the optimisation object itself. It contains the initialisation of the density distribution.

Parameters

• constraint (object of DensityConstraint class) – The constraints for this optimization problem.

• load (object, child of the Loads class) – The loadcase(s) considered for this optimisation problem.

• fesolver (object, child of the CSCStiffnessMatrix class) – The finite element solver.

• verbose (bool, optional) – Printing itteration results.

constraint

The constraints for this optimization problem.

Type

object of DensityConstraint class

load

The loadcase(s) considered for this optimisation problem.

Type

object, child of the Loads class

fesolver

The finite element solver.

Type

object, child of the CSCStiffnessMatrix class

verbose

Printing iteration results.

Type

bool

itr

Number of iterations performed

Type

int

x

Array containing the current densities of every element.

Type

2-D array size(nely, nelx)

xold1

Flattened density distribution one iteration ago.

Type

1D array len(nelx*nely)

xold2

Flattened density distribution two iteration ago.

Type

1D array len(nelx*nely)

low

Column vector with the lower asymptotes, calculated and used in the MMA subproblem of the previous iteration.

Type

1D array len(nelx*nely)

upp

Column vector with the lower asymptotes, calculated and used in the MMA subproblem of the previous iteration.

Type

1D array len(nelx*nely)

densityfilt(rmin, filt)

Filters with a normalized convolution on the densities with a radius of rmin if:

>>> filt=='density'

Parameters

• rmin (float) – Filter size.

• filt (str) – The filter type that is selected, either ‘sensitivity’ or ‘density’.

Returns

xf – Filterd density distribution.

Return type

2-D array size(nely, nelx)

disp(x, u, lamba, ke, penal)

This function calculates displacement of the objective node and its sensitivity to the densities.

Parameters

• x (2-D array size(nely, nelx)) – Possibly filtered density distribution.

• u (1-D array size(max(edof), 1)) – Displacement of all degrees of freedom.

• lamba (2-D array size(max(edof), 1)) –

• ke (2-D array size(8, 8)) – Element stiffness matrix with full density.

• penal (float) – Material model penalisation (SIMP).

Returns

• uout (float) – Displacement objective.

• duout (2-D array size(nely, nelx)) – Displacement objective sensitivity to density changes.

iter(penal, rmin, filt)

This function performs one iteration of the topology optimisation problem. It

• loads the constraints,

• calculates the stiffness matrices,

• executes the density filter,

• executes the FEA solver,

• calls upon the displacement objective and its sensitivity calculation,

• executes the sensitivity filter,

• executes the MMA update scheme,

• and finally updates density distribution (design).

Parameters

• penal (float) – Material model penalisation (SIMP).

• rmin (float) – Filter size.

• filt (str) – The filter type that is selected, either ‘sensitivity’ or ‘density’.

Returns

• change (float) – Largest difference between the new and old density distribution.

• uout (float) – Displacement at the objective node for the current design.

layout(penal, rmin, delta, loopy, filt, history=False)

Solves the topology optimisation problem by looping over the iter function.

Parameters

• penal (float) – Material model penalisation (SIMP).

• rmin (float) – Filter size.

• delta (float) – Convergence is roached when delta > change.

• loopy (int) – Amount of iteration allowed.

• filt (str) – The filter type that is selected, either ‘sensitivity’ or ‘density’.

• history (bool, optional) – Do the intermediate results need to be stored.

Returns

• xf (array size(nely, nelx)) – Density distribution resulting from the optimisation.

• xf_history (list of arrays len(iterations size(nely, nelx))) – List with the density distributions of all iterations, None when history != True.

mma(m, n, itr, xval, xmin, xmax, xold1, xold2, f0val, df0dx, fval, dfdx, low, upp, a0, a, c, d)

This function mmasub performs one MMA-iteration, aimed at solving the nonlinear programming problem:

\[\begin{split}\min & f_0(x) & + a_0z + \sum_{i=1}^m \left(c_iy_i + \frac{1}{2}d_iy_i^2\right) \\ &\text{s.t. }& f_i(x) - a_iz - y_i \leq 0 \hspace{1cm} & i \in \{1,2,\dots,m \} \\ & & x_{\min} \geq x_j \geq x_{\max} & j \in \{1,2,\dots,n \} \\ & & y_i \leq 0 & i \in \{1,2,\dots,m \} \\ & & z \geq 0\end{split}\]

Parameters

• m (int) – The number of general constraints.

• n (int) – The number of variables \(x_j\).

• itr (int) – Current iteration number (=1 the first time mmasub is called).

• xval (1-D array len(n)) – Vector with the current values of the variables \(x_j\).

• xmin (1-D array len(n)) – Vector with the lower bounds for the variables \(x_j\).

• xmax (1-D array len(n)) – Vector with the upper bounds for the variables \(x_j\).

• xold1 (1-D array len (n)) – xval, one iteration ago when iter>1, zero othewise.

• xold2 (1-D array len (n)) – xval, two iteration ago when iter>2, zero othewise.

• f0val (float) – The value of the objective function \(f_0\) at xval.

• df0dx (1-D array len(n)) – Vector with the derivatives of the objective function \(f_0\) with respect to the variables \(x_j\), calculated at xval.

• fval (1-D array len(m)) – Vector with the values of the constraint functions \(f_i\), calculated at xval.

• dfdx (2-D array size(m x n)) – (m x n)-matrix with the derivatives of the constraint functions \(f_i\). with respect to the variables \(x_j\), calculated at xval.

• low (1-D array len(n)) – Vector with the lower asymptotes from the previous iteration (provided that iter>1).

• upp (1-D array len(n)) – Vector with the upper asymptotes from the previous iteration (provided that iter>1).

• a0 (float) – The constants \(a_0\) in the term \(a_0 z\).

• a (1-D array len(m)) – Vector with the constants \(a_i1 in the terms :math:\).

• c (1-D array len(m)) – Vector with the constants \(c_i\) in the terms \(c_i*y_i\).

• d (1-D array len(m)) – Vector with the constants \(d_i\) in the terms \(0.5d_i (y_i)^2\).

Returns

• xmma (1-D array len(n)) – Column vector with the optimal values of the variables \(x_j\) in the current MMA subproblem.

• low (1-D array len(n)) – Column vector with the lower asymptotes, calculated and used in the current MMA subproblem.

• upp (1-D array len(n)) – Column vector with the upper asymptotes, calculated and used in the current MMA subproblem.

• Version September 2007 (and a small change August 2008)

• Krister Svanberg <krille@math.kth.se>

• Department of Mathematics KTH, SE-10044 Stockholm, Sweden.

• Translated to python 3 by A.J.J. Lagerweij TU Delft June 2018

sensitivityfilt(x, rmin, duout, filt)

Filters with a normalized convolution on the sensitivity with a radius of rmin if:

>>> filt=='sensitivity'

Parameters

• x (2-D array size(nely, nelx)) – Current density ditribution.

• duout (2-D array size(nely, nelx) – Displacement objective sensitivity to density changes.

• rmin (float) – Filter size.

• filt (str) – The filter type that is selected, either ‘sensitivity’ or ‘density’.

Returns

duoutf – Filterd sensitivity distribution.

Return type

2-D array size(nely, nelx)

solvemma(m, n, epsimin, low, upp, alfa, beta, p0, q0, P, Q, a0, a, b, c, d)

This function solves the MMA subproblem with a primal-dual Newton method.

\[\begin{split}\min &\sum_{j-1}^n& \left( \frac{p_{0j}^{(k)}}{U_j^{(k)} - x_j} + \frac{q_{0j}^{(k)}}{x_j - L_j^{(k)}} \right) + a_0z + \sum_{i=1}^m \left(c_iy_i + \frac{1}{2}d_iy_i^2\right) \\ &\text{s.t. }& \sum_{j-1}^n \left(\frac{p_{ij}^{(k)}}{U_j^{(k)} - x_j} + \frac{q_{ij}^{(k)}}{x_j - L_j^{(k)}} \right) - a_iz - y_i \leq b_i \\ & & \alpha_j \geq x_j \geq \beta_j\\ & & z \geq 0\end{split}\]

Returns

x – Column vector with the optimal values of the variables x_j in the current MMA subproblem.

Return type

1-D array len(n)

Plotting Module

Plotting the simulated TopOpt geometry with boundary conditions and loads.

Bram Lagerweij Aerospace Structures and Materials Department TU Delft 2018

class src_Actuator.plotting.Plot(load, title=None)

This class contains functions that allows the visualisation of the TopOpt algorithm. It can print the density distribution, the boundary conditions and the forces.

Parameters

• load (object, child of the Loads class) – The loadcase(s) considered for this optimisation problem.

• title (str) – Title of the plot if required.

nelx

Number of elements in x direction.

Type

int

nely

Number of elements in y direction.

Type

int

fig

An empty figure of size nelx/10 and nely/10*1.2 inch.

Type

matplotlib.pyplot figure

ax

The axis system that belongs to fig.

Type

matplotlib.pyplot axis

images

This list contains all density distributions that need to be plotted.

Type

1-D list with imshow objects

add(x, animated=False)

Adding a plot of the density distribution to the figure.

Parameters

• x (2-D array size(nely, nelx)) – The density distribution.

• animated (bool, optional) – An animated figure is generated when history = True.

boundary(load)

Plotting the boundary conditions.

Parameters

load (object, child of the Loads class) – The loadcase(s) considered for this optimisation problem.

loading(load)

Plotting the loading conditions.

Parameters

load (object, child of the Loads class) – The loadcase(s) considered for this optimisation problem.

save(filename, fps=10)

Saving an plot in svg or mp4 format, depending on the length of the images list. The FasterFFMpegWriter is used when videos are generated. These videos are encoded with a hardware accelerated h264 codec with the .mp4 file format. Other codecs and encoders can be set within the function itself.

Parameters

• filename (str) – Name of the file, excluding the file extension.

• fps (int, optional) – Amount of frames per second if the plots are animations.

show()

Showing the plot in a window.

class src_Actuator.plotting.FasterFFMpegWriter(**kwargs)

Bases: matplotlib.animation.FFMpegWriter

FFMpeg-pipe writer bypassing figure.savefig. To improof saving speed

classmethod bin_path()

Return the binary path to the commandline tool used by a specific subclass. This is a class method so that the tool can be looked for before making a particular MovieWriter subclass available.

cleanup()

Clean-up and collect the process used to write the movie file.

finish()

Finish any processing for writing the movie.

frame_size

A tuple (width, height) in pixels of a movie frame.

grab_frame(**savefig_kwargs)

Grab the image information from the figure and save as a movie frame.

Doesn’t use savefig to be faster: savefig_kwargs will be ignored.

classmethod isAvailable()

Check to see if a MovieWriter subclass is actually available.

saving(fig, outfile, dpi, *args, **kwargs)

Context manager to facilitate writing the movie file.

*args, **kw are any parameters that should be passed to setup.

setup(fig, outfile, dpi=None)

Perform setup for writing the movie file.

Parameters

• fig (~matplotlib.figure.Figure) – The figure object that contains the information for frames

• outfile (str) – The filename of the resulting movie file

• dpi (int, optional) – The DPI (or resolution) for the file. This controls the size in pixels of the resulting movie file. Default is fig.dpi.