2. ME 6607 – Optimal Design of Multi-functional Systems
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Problem Statement
The structure: A MBB structure is considered with one load at the middle point in its top surface, as shown in Table 4, middle row in the
paper. The load can be considered as a unit load. The material has a modulus of elasticity of 100 MPa (or the unity) and the Poisson’s
ratio of 0.3. Set L equal to 1.0 cm.
Part 1: Conduct the topology optimization of the MBB structure using the 3D SIMP method for the minimization of the strain energy
function c as defined in the code. For this project, use the parameters listed in Table 4, middle row. Conduct the topology optimization
of the MBB structure for the following cases of filters:
Density filter only;
Sensitivity filter only;
Grey scale filter only;
Density filter and grey scale filter;
Sensitivity filter and grey scale filter.
Among the five cases, for this MBB structure, indicate which case provides the most black-and-white solution.
Part 2: Iterative Solver: (Section 6.1.5)
Conduct the topology optimization of the MBB structure, using the grey scale filter only, for the following cases of mesh size:
30 x 5 x 5
60 x 10 x 10
120 x 20 x 20
Provide the comparison of the two methods, the direct solver and the iterative solver, in a table similar to Table 3, to show the effect
of the different solvers.
3. ME 6607 – Optimal Design of Multi-functional Systems
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Solution Methodology:
The given code was analysed and the functioning of the various scripts was studied. The problem was studied so as to implement the
loads and boundary conditions and different filters. The Matlab code from the source was used for implementation of the program for
minimising the value of objective function so as to find the optimal design.
Results & Discussion:
The results of various cases and the plots of optimal designs are attached as an attachment to the file. The optimal designs are found
to be dependent on filter size “penal, filters, solvers and mesh sizes” and it also requires measures for scaling, stability and convergence
for larger values of Young’s modulus FEM mesh resolution drastically changes the results without suitable filters and it is left to the
discretion of the end user. A much finer mesh gives more details on the structure.
Ref. M. P. Bendsoe, O. Sigmund -- Topology Optimization -- Theory, Methods and Applications -- ISBN 3-540-42992-i Springer-Verlag Berlin
Heidelberg New York, www.top3dapp.com
The plots of optimal design for various cases are shown below:
Fig. 1 Density Filter optimized structure Fig. 2 Sensitivity Filter optimized structure
Fig. 3 Grayscale Filter optimized structure Fig. 4 Density & Grayscale Filter optimized structure
Fig. 5 Sensitivity & Grayscale Filter optimized structure
4. ME 6607 – Optimal Design of Multi-functional Systems
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Comparison of Time usage for Finite Element Analysis of different solvers
Mesh Size Direct Solver Iterative Solver Remarks
30 x 5 x 5 0.028s 0.45s As the mesh size increases
the performance of the
iterative solver is better than
the direct solver.
60 x 10 x 10 0.65s 1.6s
120 x 20 x 20 34s 26s
Observations:
It is to be studied whether usage of a filter size equal to element size has a better convergence and solution when compared to other
filter sizes. The results of those tests are also attached with this for supporting this argument. When filter size is equal to element size
the filter produces almost identical results irrespective of filter used.
