# Topology And Shape Optimization Of Design Space Engineering Essay

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This report describes how a, 2D parametric optimization approach is used for a given design space for a bracket under a load of 25 N, with objective to find an optimum shape within the geometrical constraint. The methodology is to first find a design layout using topology optimization with decrease volume of 50%. Design parameters were assigned to geometry obtained from topology optimization. Topology optimization addresses an initial possible geometry within the design space. Design sensitivity analysis was carried out to check the response of the state variable (max von misses stress) and objective function (volume) by changing the design variables that define the shape of the layout. Shape optimization was used to minimize the volume as an objective function of the initial design geometry by optimizing parametrically different design variable under constraint of maximum von Mises stress as a state variable. Two Ansys in-built methods for parametric optimization, the sub-problem approximation and, first order method, were used as optimization tool and finally there results were compared. The initial volume of 40,000 mm3 was optimized to 12310 mm3 and 12310 mm3 by using sub-problem approximation and first order optimization.

Table of content

Introduction

In this project we have studied how to use design optimization approach for a given design space, to get an optimum shape within the design constraints of the given space and constrain of max von misses stresses. The given design space as shown in Figure 1, was optimized by using two concepts of optimization; topology optimization and shape optimization. The design layout obtained from topology optimization was optimized in shape optimization. The objective was to obtain an optimum shape with a decrease volume/weight. In topology optimization the volume of the given design space were decreased by 50 % for a load of 25 N.

Figure 1 Design domain and required optimum shape.

## Motivation:

This is the decade of economical disaster for the industry. It is important to reduce the cost of product for survival. Time is also a very important factor in every aspect of life especially in industry. In my view cost of the product strongly depend on time. How one can reduce net cost of the product. This thing brings my attention to do work on it. As the title (Topology and shape optimization) of our report shows Former reduces the volume maintaining the same stiffness and the later gives us the approximated shape of the product. It fulfills our objective to reduce the cost of the product that is the need of the today's industry.

The objective of this project was

How to use Ansys in-built optimization tool for an arbitrary design space, as in this project a simple design space were considered for a loading shown in Figure 1.

Which optimization tool is more efficient (less computational time) and accurate?

Sensitivity analysis, which design parameters are more sensitive towards objective function (volume) and state variable (max von Mises)

Throughout the entire project Ansys classic was used. The report consists of how the structural layout was obtained from the design space using topology optimization, how different design parameters were optimized using shape optimization. Probabilistic design analysis was carried out to find sensitivity of design parameters and finally the conclusion and discussions are presented.

## Problem solution

The given problem is analyzed with the steps as shown in flow diagram in Figure 2. During the whole optimization high strength steel (E = 200 MPa, ν = 0.3) was used. The yield strength of the material was assumed to be MPa and allowable stress were assumed to be

MPa,

Where SF=factor of safety= 1.5 finally,

Assumed fatigue limit = 450 MPa.

Figure 2 Flow diagram of Optimization.

Modeling:

Initial design layout

In order to find an initial geometry within the design space topology optimization was used, which is described below.

Topology Optimization

In topology optimization the structural geometry is described by material distribution that varies from 0 to 1 in a given design space. In finite element analysis the material distribution can be interpreted as the element pseudo-density distribution. The goal of the topology optimization is to find the pseudo-density distribution, which minimizes the compliance. In this method half symmetry was considered because of symmetric loading. The known design domain was discretized with finite element mesh as shown in Figure 3. Thereafter, the loads and boundary conditions were applied. The design objective was to decrease the volume within the design space by 50 %. Initial volume of domain was 40,000 mm3. The model was meshed with equal element length in both x and y direction. The element used in Ansys was PLANE82 which is a higher version of 2D-element having two degrees of freedom at each node, translation in nodal x and y directions. Symmetric boundary condition was used at the symmetric line. Topology optimization was run in Ansys for 50 iterations and the approach was taken optimally. The density plot shown in Figure 4 was geometry obtained for volume reduction of 50 %.

Figure 3 Finite element mesh of the design domain.

Figure 4 Resulting density distribution after topology optimization.

Since the geometry in topology optimization was meshed by using mapped meshing, it is easier to find the optimized geometry from the density distribution shown in Figure 4, the density distribution shown as blue in Figure 4 is of no interest so the optimized geometry is shown in figure 5, whose dimensions are,

Where

Figure 5 Density distribution of initial geometry.

## Fatigue criterion

Before the optimization of the two layouts, the two geometries were first analyzed for static load of 25 N, to check whether fatigue analysis will be required or not. The criterion for fatigue was:

If, then fatigue will occur and

If max , then there will not be any fatigue.

Initial stress analysis

The stress distribution is shown in the Figure 7.

Figure 7 Initial von Mises stress distribution for layout 1.

Here the max von Mises stress of 508.69 MPa is due to sharp corner, so the geometry was modified by using fillet. Also L1 was increased from 25 mm to 30 mm and thickness T1 was decreased to from 15 mm to 10 mm to accommodate the fillet in the design space. The modified geometry and dimensions are shown in Figure 8.

Figure 8 Modified geometry of the layout.

The new geometry was again analyzed for the static load 25 N, the maximum von Mises stress was now 423.69 MPa as shown in Figure 9.

Figure 9 Modified mesh and stress distribution of layout.

Since the maximum von Mises stress is below the fatigue limit, any further analysis of fatigue will not be necessary.

## Sensitivity analysis

Before design optimization Ansys probabilistic design module was used to find the sensitivity of design variables to state variable and objective function. A truncated Gaussian approach was used. Sensitivity analysis is used to determine how sensitive a model is to changes in the values of the design parameters of the model.

Sensitivity analysis for layout

The sensitivity plot of design variables for state variable Smax is shown in figure 23.

Figure 23 sensitivity plot of design variables for Smax for layout.

This shows that maximum von Mises stress in more sensitive to design variables H1, H2, H3, R1 and L1. For objective function (volume) the sensitivity plot is shown in Figure 24.

Figure 24 sensitivity plot of design variables for Volume for layout.

The sensitivity plot for the objective function shows that design variables T2, H1, H2, L1, ANG_1 and ANG_2 are more sensitive to be optimized to decrease volume of geometry.

## Shape Optimization

The optimization problem consists of iterations to find the vector of the design variables that give the minimum of the objective function while meeting all limitations on the design and state variables. The layout obtained from topology optimization was optimized by two Ansys in-built methods of optimization, namely

Subproblem approximation ,

First order optimization.

First, Mathematical model for layout is presented.

Mathematical model

The mathematical model of optimization is presented by state variables, objective function and design variables.

State variable.

The functional state of the design is characterized by state variables. For example, during analysis the maximum von Mises stresses in any element must not be greater than the permissible stress, which is set to be 600 MPa.

The limit on the state variable

Where

MPa

Objective function:

f =f(x) = volume

Design variables.

The variable parameters are presented in the form of vector of the design variables:

Mathematical model for the layout

State variable

Where for

MPa

Objective function:

f =f(x) = volume

Design variables.

Vector of design variables is

The limits on design variables:

Subproblem Approximation

The subproblem approximation method relates to zero-order methods where calculation of derivatives of dependent variables (objective function and state variables) are not required and only the values of the dependent variables, which are initially approximated by approximate dependence with the help of the least-squares method are calculated. Subproblem approximation was used for both cases whose results are presented below.

Subproblem approximation for layout

In this method the initial volume 19439.5 mm3 was decreased to 12310 mm3. The problem was run for 50 iterations, and was converged at iteration no 41. All the design sets are shown in [Appendix 1]. A total processing time of 66 seconds was recorded. Figure 11 shows stress distribution before and after optimization; similarly figure 12 and 13 shows history of objective function, state variable and design. Table 1 shows optimized and initial parameters.

Figure 11 von Mises stress before and after optimization for layout 1 using subproblem approximation.

Figure 12 Evolvement of the objective function and state variable for layout 1 using subproblem optimization.

Figure 13 Evolvement of the design variables for Subproblem optimization

Table 1 Subproblem approximation for layout 1.

parameters

Variables

optimized values

Initial values

SMAX

(SV)

603.88 MPa

423.69 MPa

ANG_1

(DV)

73.688°

74°

ANG_2

(DV)

70.706°

71.5°

H1

(DV)

48.99 mm

50 mm

H2

(DV)

40.04 mm

40 mm

H3

(DV)

31.88 mm

26 mm

R1

(DV)

4.84 mm

5 mm

R2

(DV)

8.34 mm

10 mm

R3

(DV)

4.1 mm

5 mm

R4

(DV)

8.35 mm

10 mm

T1

(DV)

10.56 mm

10 mm

T2

(DV)

5.97 mm

12.5 mm

L1

(DV)

31.87 mm

30 mm

L2

(DV)

80.8 mm

85 mm

VOLUME

(OBJ)

12310mm3

19439.5 mm3

First order optimization

The first order method converts the problem to an unconstrained one by adding penalty functions to the objective function. In this method the actual finite element representation is minimized and not an approximation. The first order method uses gradients of the dependent variables with respect to the design variables, gradient calculations are performed for each iteration in order to determine a search direction, and a line search strategy is adopted to minimize the unconstrained problem. Thus each iteration is composed of a number of sub iterations that include search direction and gradient computations. That is why one optimization iteration for the first order method performs several analysis loops.

First order optimization for layout

In this method the initial volume 19439.5 mm3 was decreased to 13045 mm3. The problem was completed for 20 iterations, but was converged at iteration no 6. All the design steps are shown in [Appendix 3]. A total processing time of 105 seconds was recorded. Figure 17 show stress distribution before and after optimization, similarly Figure 18 and 19 shows evolvement of the objective function, state variable and design against no of iterations. Table 3 shows optimized and initial parameters.

Figure 17 von Mises stress before and after optimization for layout 1 using first order optimization.

Figure 18 Evolvement of the objective function and state variable for layout 1 using first order optimization.

Figure 19 Evolvement of the design variables for layout 1 using first order Optimization.

Table 3 First order optimization for layout

Parameters

Variables

optimized values

Initial values

SMAX

(SV)

595.06 MPa

423.69 MPa

ANG_1

(DV)

74.8°

74°

ANG_2

(DV)

70°

71.5°

H1

(DV)

49.1 mm

50 mm

H2

(DV)

40.41 mm

40 mm

H3

(DV)

28.22 mm

26 mm

R1

(DV)

4.99 mm

5 mm

R2

(DV)

10.01 mm

10 mm

R3

(DV)

5.03 mm

5 mm

R4

(DV)

9.86 mm

10 mm

T1

(DV)

9.22 mm

10 mm

T2

(DV)

8.04 mm

12.5 mm

L1

(DV)

31.11 mm

30 mm

L2

(DV)

75.00 mm

85 mm

VOLUME

(OBJ)

13045 mm3

19439.5 mm3

## Conclusions and discussion

From this analysis it was concluded that,

Sensitivity analysis should be performed before optimization to find out which design parameter is more sensitive toward state variable and objective function. It will decrease the cost processing.

2D optimization should perform before 3D optimization to get initial optimum design.

Sub problem approximation method is faster than first order optimization.

Sub problem approximation is more sensitive to meshing, because of high variation in design variables at each iteration,

Less infeasible solution is obtained through first order optimization

First order optimization method is more suitable for problem with high exactitude.

## References

[] ANSYS Inc. Ansys academic research, help system, Release 11.0.

[2] http://www.feadomain.com