Computer exercise in structural optimization: shape and ... - math

Computer exercise in structural optimization: shape and topology optimization using TRINITAS Introduction to Structural and Multidisciplinary Optimization (TMV 145) 04/05

1 Introduction

In this exercise, you are going to try to describe the shape using both simple straight lines and cubic B´ezier splines. When making a model of the structure shown in Fig. 1, you should take advantage of the fact that the structure and loading is symmetric, i.e. make a model of the right (or left) half of the structure only, and use appropriate (symmetry-imposing) boundary conditions.

The objective of this small computer exercise is to provide a hands on experience with structural optimization. We suggest you to solve some simple yet realistic mechanical problems. The problems are to be solved using TRINITAS. The usage of TRINITAS for shape optimization is described in P. W. Christensen “Shape Optimization with TRINITAS”. Be aware that the current version of TRINITAS is somewhat newer than the one described in the manual by Christensen and that there therefore are quite a few discrepancies. The results of the exercise are to be described in a short report. You can download TRINITAS at http://www.solid.ikp.liu.se/∼botor/Prog/Executables.html.

Load A

B

2L

φ

L

2L

2 Part I: Shape optimization 2.1 Problem formulation The shape that minimizes the weight of the road pole in Fig. 1 is to be found. You may change the shape of all parts of the road pole in any way, as long as the design constraints 1–3 below are fulfilled. This means for instance, that it is allowed to move the “points” where the road pole meets the soil. The design constraints are that: 1. The maximum von Mises stress must not increase compared to the original design in Fig. 1. 2. The line between points A and B is not allowed to change in length or position, i.e. the height and width of the road pole must not change compared to the original design. 3. The angle φ must be 90◦ . To perform shape optimization, the shape of the road pole must be described in terms of the design variables.

L

L

L

L

Figure 1: Original design of the road pole; a car passing the bridge would drive into the paper.

2.2 Modeling the road pole As mentioned above, you should try two different representations of the part of the boundary that is allowed to change: (i) describe the boundary with cubic B´ezier splines, (ii) describe the boundary with simple straight lines.

1

. You can modify your mesh (e.g., increase the number of elements) via → → → . . . (Hint: the number of elements in the mesh is displayed with the menu command → → → but create your mesh first!.) • Avoid using design spans where the design point reaches one of the end points of the span, i.e. do not use too short spans. • Try different span directions and meshes. • Save the initial design and all optimum designs. (Hint: use the menu command → .) NOTE: T HE INITIAL DESIGN MODEL WILL BE USED IN THE FOLLOWING PART OF THE LAB , SO MAKE SURE YOU SAVE IT PROPERLY !

As a suggestion the circular arc in Fig. 1 could be approximated using four lines/splines, see Fig. 2.

Working Plane 1

Figure 2: Two different approximations of the circular arc; straight lines to the left and B´ezier splines to the right. The magnitude of the load on the line A-B is of no importance, as the response of the structure is linear with respect to the load. The load resulting from the structure’s own weight can be neglected (assume for instance that the structure is a miniature road pole!). Some advice: • To describe a geometry in TRINITAS proceed as follows. First, create points (some of which might later become your design points): → → . Then, join them with straight lines or B´ezier curves as needed → . After you have created a 2D shape, you must specify that it is in fact a surface of a 3D object; you do so via → and clicking on all lines bounding a surface. Finally, create a volume with constant thickness ( → → and click on the surface). After that, apply the appropriate boundary conditions ( → → , or , whatever feels more appropriate; also → or ). • Create PostScript screen-shots of TRINITAS screen by using → . TRINITAS then silently creates a new file postscripti.ps in the current working directory. • Use a free mesh with some 300–400 quadratic triangular elements → →

2.3 Optimizing the road pole The weight of the structure should be minimized. As weight minimization is not implemented in TRINITAS, use volume minimization instead. (Volume minimization is equal to weight minimization in this case, because the density of the road pole is constant.) The stresses (von Mises) should not exceed the maximum stress in the initial design shown in Fig. 1. To ensure that, perform a linear stress analysis of the structure, read the maximum von Mises stress from the screen, and introduce the appropriate stress constraints for 2D shape minimization. When only straight lines are used to describe the boundary, no additional geometrical constraints should be used. However, when splines are used, the boundary should be constrained to be C1 -continuous (except for the 6 corners). (Do not forget that you have modeled only one half of the structure!) Also, the angle φ in Fig. 1 should always be 90◦ . Hints: • The main optimization menu can be found under: → → . • The menus for adding the C1 -continuity constraints can be found under → → → → . • Weight and volume calculation is enabled with the menu command → → → .

2

Consider the aircraft floor support beam in Fig. 3, and suppose the domain should be filled to 50%. This example is frequently referred to as the MBB-beam, after the Munich company Messerschmitt-B¨olkow-Blom where the example was initially proposed. Fig. 4 shows numerical solutions obtained using topology optimization Figure 4: Topology optimization solution for 150 × 50 method with SIMP. mesh for the MBB-beam. Only the right half is shown.

3.2 Topology optimization TRINITAS and → →

menus

in

→ .

The main optimization menu for topology optimization can be found under: → → . should typically be kept above 10, and For simplicity only stiffness maximization is considbelow 1E − 3. ered. This means that we like to minimize the compli and are reance ℓ(u(ρ )), where u(ρ ) is the solution of the equiρ and V in (1) according to lated to the parameters librium equation(s) for the thickness ρ (in 2D); see the hand-out “Topology optimization by distribution of · |Ω| = V, (2) isotropic material.” The design constraints on ρ include: where |Ω| is the area of Ω, and

ρ ≤ ρ (x) ≤ ρ ,

Z

Ω

ρ (x) dx ≤ V,

· = ρ . (3)

(1)

where Ω is the (plane) domain to be covered by finite From (2) and (3) it follows that the relative volume fracelements, and V is the available volume of elastic ma- tion is terial. The ρ -function is always assumed to be approxV 1 = . imated as constant in each finite element. TRINITAS ρ · |Ω| also uses the SIMP material interpolation law (with TRINITAS’ default values are = some exponent p); again, see the cited hand-out. 1 (found under → → ) and = 3. 3.1 Topology optimization in 2D This means that by default V /(|Ω|ρ ) = 1/3, i.e. the Using the term “topology optimization”, one often “admissible box” should be filled with material to no refers to the problem where, in addition to (1), ρ is re- more than one third. is the exponent p in quested to take (almost) exclusively the bounding values ρ and ρ in the entire domain Ω. Since algorithms the material interpolation law SIMP. The value 1 correfor discrete values of the optimization variables are not sponds to “no penalization”. Use values in the range as developed, the request for discrete values is in topol- 2, . . . , 3 to obtain reasonable designs when ρ should ogy optimization typically met by penalizing the inter- take (almost) only the values ρ and ρ . is (proportional to) the compliance ℓ(u(ρ )). when choosing p > 1 in the SIMP method. TRINITAS uses the OC-algorithm, which is a lin0.1 earization in the intervening variables ρi−a (where ρi is the thickness in element i). The value a is related to through 1

= 1/(1 + a).

(4)

0.05

Normally, values for the lie between 0.5 and 0.8, where the lower ones seem suitable for higher SIMP exponent values and vice versa.

6

Figure 3: Geometry and load case for the MBB-beam. 3

3.3 Solving the MBB-beam example

Volume Exponent>, starting solution and number of iterations.

Consider a load and geometry similar to the case shown in Fig. 3. By using symmetry arguments only one half needs to be used in the computations, as shown in 3.4 Design of a road pole, revisited Fig. 5. (For simplicity point load and support is used Consider again the road pole from the first computer project, see Fig. 1. Suppose that the manufacturing requirements are such that one needs constant thickness of the poles. The idea is that by changing the topology, in addition to the shape, maybe we can find an even better road pole. Use the amount of volume you arrived at in the first 2 project, and then perform a stiffness maximization with this correct value of V . You can use the geometry definitions from the shape optimization, i.e. half the domain 5 shown in Fig. 1. Do you get a suggestion of any new boundaries/holes? Where? Are some of the boundaries simFigure 5: Half of the MBB-beam to be used in the comilar in shape and position to those obtained after the putations. shape optimization in the first part of the lab? instead). Solve this topology optimization problem with 4 Report TRINITAS if the domain should be filled to one third! Use approximately 35 × 14 Lagrange quadratic ele- You should write a report containing: ments. You start and restart the algorithm via . Can you get a result similar to the design • Plots of all relevant optimal designs, with numeridepicted in Fig. 4? You should describe how you have cal data and short comments. chosen the values of ,