Topology and shape optimization for non-linear problems. Edward de Boer MT Sydney, December 2010

Transcription

1 Topology and shape optimization for non-linear problems Edward de Boer MT 11.1 Sydney, December 21

2 Contents 1 Introduction 2 2 Bi-directional evolutionary structural optimization 3 3 Topology optimization of energy-absorbing structures Objective of the optimization for energy-absorbing structures Sensitivity number at end displacement Sensitivity number for whole displacement history Evolutionary history of sensitivity number Checker-boarding suppression algorithm Material removal/addition criterion Implementation using Abaqus Examples Example Example Inverse shape optimization Objective of the optimization method Evolutionary history of indicator number Material removal/addition criterion Implementation Examples Example Example Example Example Conclusion 34 A Python code 37 A.1 Topology optimization of energy-absorbing structures A.2 Inverse shape optimization

3 Chapter 1 Introduction A stent is a mesh-like, dilatable structure which is inserted into various natural passages/conduits in the human body to counteract a disease-induced flow constriction. In order to insert a stent into the human body, it is only dilated (by means of inflation) after insertion. During inflation, the diameter of the stent is typically increased by a factor of two. The shape of the deformed stent is critical to it s ability to function. Therefore there is need for an algorithm to optimize the final shape of the stent. In this report, a simple algorithm is developed to optimize the initial shape of a material such that under large deformation, the material deforms to a desired shape (so called inverse shape optimization). It is assumed that the design criteria for the stent can be translated into a combination of simple shapes such as circular and rectangular holes in a planar design domain. This way the algorithm can be used on a thin-walled cylinder to optimize the shape of a stent. The algorithm that is developed, is based on the bi-directional evolutionary structural optimization (BESO) method. The BESO method has been developed for the topology optimization of structures in for example automotive. BESO has received a lot of attention because it can be easily implemented and linked to existing finite element analysis (FEA) packages. The BESO method is further explained in chapter 2. In order to get familiar with the BESO method and to serve as a benchmark, two examples from a paper by Huang, Xie and Lu [4] on topology optimization of energy-absorbing structures are reproduced in chapter 3. A combination of Abaqus with the python interface is used to do the optimization. Finally the inverse shape optimization method is introduced in chapter 4. Some examples demonstrate the functionality and limitations of the algorithm under large deformation compression and tension. A non-linear elasto-plastic material is used for all examples. 2

4 Chapter 2 Bi-directional evolutionary structural optimization Research in structural topology optimization has lead to the development of the evolutionary structural optimization [2] (ESO) method and as an extension to that, to the bi-directional evolutionary structural optimization [3, 5] (BESO) method. Both methods have received a lot of attention [7, 6, 8, 9, 1] because they are easily implemented and linked to finite element analysis (FEA) packages. In the ESO process, the design domain is discretized in elements. From a full design domain, redundant elements are then removed in order to achieve an optimal design. The optimum solution can obviously not be achieved in one step. Therefore an evolutionary procedure has to be adopted. Only a small number of elements is removed in every iteration step. The number of elements that is to be removed each iteration is prescribed by the evolutionary ratio (ER). The evolutionary ratio is the number of elements to be removed divided by the total number of elements in the initial design. ESO is limited by the fact that removed elements cannot be added back to the design. To overcome this, the BESO (bi-directional evolutionary structural optimization) method was developed. In the BESO method, elements can be either added or removed. The number of elements that is removed or added is determined by the evolutionary ratio (ER) and the admission volume ratio (AR). The admission volume ratio is the number of added elements divided by the total number of elements in the initial design. Usually one of the optimization criteria is a volume constraint. The BESO algorithm continues until the target volume is reached. In order to determine what elements should be removed or added, sensitivity numbers are used. The sensitivity numbers can be derived mathematically by differentiating the objective to the design variable. Present elements that are least favorable to the objective are removed, while void elements that will be most favorable to the objective are added to the design. 3

5 Chapter 3 Topology optimization of energy-absorbing structures Energy absorbing structures are useful in automotive, where collision may cause severe injuries to vehicle occupants. In order to prevent injuries, there are several constraints on the energy absorbing structure. A structure that is too stiff is undesirable because the forces in that structure could exceed the maximum tolerable crushing force. A too flexible structure is also undesirable because it may have a long crushing distance, and therefore some parts of the structure may intrude and penetrate the human compartment. The goal of the presented optimization method is to design a structure that operates within the prescribed limits of force and displacement, and can absorb as much energy per unit of volume as possible. Max. allowable crushing force F* Stiff structure Feasible optimal structure Max. crushing distance Flexible structure Crushing distance U* Figure 3.1: Load-displacement curve for three typical energy absorbing structures. 4

6 3.1 Objective of the optimization for energy-absorbing structures The objective is defined as Maximize x subject to f(x) = E V F max = F U max = U x j {, 1}, j = 1,..., M. where E is the strain energy, V is the (undeformed) volume of the structure, F is the external force and U is the displacement. F and U are threshold values for the force and displacement respectively. The design variable x j declares the presence (1) or absence () of an element. M is the total number of elements in the design domain. Typically, a non-linear material must be considered. The crush behavior is simulated by gradually increasing the applied displacement up to the maximum allowable crushing displacement (U max ). In every displacement step, the equilibrium equation must be satisfied (3.1) R i = F i F int i = (3.2) where R i is the residual force at displacement step i, F i is the external force vector at step i and Fi int is the internal force vector at step i. The internal force vector can be expressed as F int i = M J=1 C jt F j i (3.3) where C jt represents a matrix that transforms the nodal internal force vector for the jth element into a global internal force vector Sensitivity number at end displacement In order to find the optimal design, the derivative of the objective is evaluated. df(x) dx = 1 V de dx E V 2 dv dx According to the principle of energy conservation, the strain energy is equal to the applied work (see figure 3.2). The strain energy can be expressed as 1 E = W = lim n 2 (3.4) n [(Ui T Ui 1)(F T i + F i 1 )] (3.5) i=1 where n is the total number of displacement steps. To calculate the variation in strain energy, a new equation is introduced by adding a series of Lagrangian multipliers. 1 E = W = lim n 2 n [(Ui T Ui 1)(F T i + F i 1 ) λ T i (R i + R i 1 )] (3.6) i=1 Because the residual forces are equal to zero, the modified equation (3.6) is still the same as equation (3.5). The variation of strain energy now becomes de dx = lim 1 n 2 n [(Ui T i=1 Ui 1)( T df i dx + df i 1 T dx ) + (du i dx du i 1 T dx )(F i + F i 1 ) λ T i ( dr i dx + dr i 1 dx )] (3.7) 5

7 Before element removal F i F i-1 After element removal U i-1 U i Displacement U n Figure 3.2: Force-displacement curves before and after element removal. At the nodes where the displacement is prescribed, du dx = (the displacement will not vary over the design variable) and at all other nodes there is no external force (F = ). Therefore the second set of terms is equal to zero. For a small loading step, a linear force-displacement relationship can be assumed. F i F i 1 = K t i (U i U i 1 ) (3.8) Therefore the variation of the external force can be approximated by And so df i dx = δf i δx + δu i Kt i δx Kt i df i 1 dx = δf i 1 δx Kt i δu i δx + Kt i δu i 1 δx δu i 1 δx (3.9) (3.1) df i dx + df i 1 dx In a similar fashion it can be shown that dr i dx + dr i 1 dx = δf i δx + δf i 1 δx = δr i δx + δr i 1 δx The above equations can be substituted into equation (3.7). (3.11) (3.12) de dx = lim 1 n 2 n [(Ui T i=1 Ui 1)( T δf i δx + δf i 1 δx ) λt i ( δf i δx δf int i δx + δf i 1 δx δf int i 1 )] (3.13) δx where δf i δx and δf i 1 δx are the only remaining unknowns. The variable λ i is chosen such that these unknowns are eliminated. λ i = U i U i 1 (3.14) 6

8 By substituting λ i into equation (3.13), the variation of external work can be written as de dx = lim 1 n 2 n [(Ui T i=1 U T i 1)( δf i i nt δx δf int i 1 + )] (3.15) δx In this optimization method, the topology can evolve by the removal or addition of elements. The variation in the objective, energy and volume upon element removal can be expressed as f(x) = 1 V ( E E) V V (3.16) 1 n E = lim [(U n i T U T 2 i 1)( Fi i nt + Fi 1)] int (3.17) i=1 V = V j (3.18) where V j is the volume of the jth element. By rewriting equation (3.16) to a dimensionless form, the sensitivity number at end displacement can be expressed as αn j = V j V Ej n E (3.19) The sensitivity number can have a positive or a negative value, which indicates that the objective function may decrease or increase upon element removal or addition. To maximize the objective function, solid elements (x j = 1) with the highest positive sensitivity number should be removed while void elements (x j = ) with the lowest values should be added to the structure Sensitivity number for whole displacement history The effects of removing an element on the objective will be different for various crushing distances. In order to achieve a high efficiency for absorbing energy over the whole displacement history, the sensitivity number for the whole displacement history is defined. α j = n α j i (3.2) i=1 In this chapter the crushing displacement will be gradually increased from to U max in 1 steps. The sensitivities will be calculated in every step. By using equation (3.2) for the calculation of the sensitivity number, the final topology will have a high efficiency for absorbing energy over the whole displacement history Evolutionary history of sensitivity number To further increase the stability of the optimization process, the sensitivity number can be averaged as α j = αk j + αk 1 j 2 where αj k is the sensitivity for the current (k-th) iteration and αk 1 j previous (k 1th) iteration. (3.21) is the sensitivity in the 7

9 3.2 Checker-boarding suppression algorithm Checkerboard patterns are quite common in various fixed grid finite element based topology optimization methods. The BESO procedure is driven by elemental reference factors. In a continuum structure which is discretized using low order bilinear (2D) or trilinear (3D) elements, these reference factors could become discontinuous and result in checker-boarding as well as mesh-dependency. To prevent this, a simple checker-boarding suppression algorithm [5] will be adopted. In this method the elemental reference factors are smoothed out locally. First, the reference factors are calculated at every node by averaging the values of the elements connected to the node. e j n = M1 l=1 V le l n M1 l=1 V l (3.22) where e j n represents nodal strain energies (without any physical meaning) of the j-th node and M1 denotes the total number of elements connected to the node. V l and E l n represent the l-th element s initial volume and strain energy at the n-th increment respectively. Secondly the nodal strain energies are further averaged in order to calculate the improved elemental strain energies. E j n = M2 i=1 w(r ij)e i n M2 i=1 w(r ij) (3.23) where w(r ij ) is a weighting factor and M2 denotes the total number of nodes connected to the j-th element. The smoothing technique works as a filter based on a length-scale r min. The weighting factor is defined as w(r ij ) = r min r ij, (i = 1, 2,..., M2) (3.24) where r ij is the distance from the center of the jth element to the ith node. In effect the filter smooths out the strain energies with a radius of r min. As a rule of thumb the value for r min should be at least two times larger than the element size to prevent checker-boarding. The filter also serves as a way to extrapolate the elemental reference factor to void elements. 3.3 Material removal/addition criterion Conventional BESO methods (like in [3]) use a volume constraint. In the energy-absorption problem however there is only a constraint on the maximum crushing force. In order to satisfy that constraint, the volume is either decreased or increased. By decreasing the material volume, the part generally becomes more compliant and the maximum crushing force is consequently decreased. V k+1 = V k (1 ER) (3.25) where ER is the evolutionary volume ratio. In this chapter ER =.1 is used. Similarly if the crushing force is below the maximum allowable crushing force, it can be increased by increasing the volume. V k+1 = V k (1 + ER) (3.26) To ensure not too many elements are added in a single iteration, the admission volume ratio (AR) is introduced. The admission volume ratio is defined as the volume of added elements 8

10 divided by the total volume in the current design. The admission volume ratio must not be larger than the maximum admission volume ratio. AR AR max (3.27) where AR max =.2 is used in this chapter. The sensitivities are calculated for all elements (both solid and void) and then sorted according to their value. Void elements (x j = ) are switched on if And solid elements (x j = 1) are removed if α j < α th add (3.28) α j α th del (3.29) where αdel th and αth add are the sensitivity threshold values for the removal and addition of elements. The threshold values are determined by the following simple steps: Step 1: Let αdel th = αth add = αth. α th can be determined by V k+1. For example, if there are 4 identical elements in the design domain and α 1 < α 2... < α 4 and V k+1 corresponds to a design with 23 elements then α th = α 23. Step 2: Calculate the admission volume ratio. If AR AR max skip step 3, otherwise recalculate αdel th and αth add as in step 3. Step 3: Calculate αadd th by first sorting the sensitivity number of void elements (x j = ). The volume of elements to be switched from to 1 will be equal to AR max multiplied by the total volume in the current design. αadd th is the sensitivity number of the element ranked just above the last added element. αdel th is then determined so that the removed volume is equal to (V k+1 minus V k plus the volume of the added elements). 3.4 Implementation using Abaqus For the implementation of the mentioned optimization method, a combination of Abaqus with the python interface is used. The python script sets the initial design to a pre-made Abaqus model. Abaqus is then used to calculate forces and deformation of the model. These values are then used to calculate the sensitivity number for every element. The volume of the next design is determined by evaluating the crushing force, and the new design can be determined by evaluating the sensitivites. The new design can now be solved using FEA and the proces repeats itself untill convergence of the objective is reached. Figure 3.3 shows a flow chart of the iterative process. 9

11 Assign initial guess design Solve model using FEA Calculate sensitivities (α) j Determine volume next design (V k+1) by evaluating the maximum crushing force (F ) max No Determine admission volume ratio (AR) Determine threshold sensitivity number ( α th ) Remove and add elements to assign new design Has convergence been reached? Yes Current design (V ) is optimal k Figure 3.3: Schematic of the topology optimization method for energy-absorbing structures. In practice, removing and then adding elements back into the model is very difficult. Therefore instead of actually deleting elements, it s material properties (e.g. Young s modulus and hardening modulus) can be reduced by a factor of 1 4. The material properties of the deactivated elements will then approximately compare to the activated elements as air does to steel. The python script that is used to interface with Abaqus is included in appendix A Examples In order to test the implementation of the optimization method, the examples discussed in [4] are used as a benchmark. The example in section demonstrates how the optimization method can be used from an initial full design to find an optimal design using both sensitivities for the end displacement (criterion 1) and for the whole displacement history (criterion 2). The second example (section 3.5.2) demonstrates how an initial guess design can be used to reduce computational time in finding the optimal design. 1

12 3.5.1 Example 1 U = 2 mm 2 mm 4 mm 4 mm 1 mm Figure 3.4: Design domain, dimensions and boundary conditions for example 1. The gray area denotes the initial full design. A 1mm by 2mm beam is supported at 2 positions at the bottom (see figure 3.4) and a displacement load of U = 2mm is applied at the center of the top edge. The material is elastoplastic with Young s modulus E = 2GP a, Poisson ratio v =.3, yield stress σ y = 3MP a and plastic hardening modulus E p =.3E. The maximum allowable crushing force is set to 2kN and maximum crushing distance is set to 2mm. The design domain is discretized in node plane stress elements. The evolutionary volume ratio is set to ER =.1, the admission volume ratio is AR max =.2 and the mesh-independent filter radius is R c = 3mm. The initial design has 1% of the solid material in the design domain. The evolutionary history of the optimization using both the sensitivity for end displacement (criterion 1) and the sensitivity for the whole displacement history (criterion 2) are shown in figure 3.5. BESO searches for an optimal design by removing and adding material. In both cases the absorbed energy per unit of volume generally increases as the volume decreases. The crushing force for the initial design is larger than the maximum allowable crushing force (see figure 3.6), therefore the volume is gradually decreased in order to meet the requirement. As the topology evolves, trusses emerge and are later destroyed. These changes in topology cause bumps in the absorbed energy per unit of volume. 11

13 E/V [MN/M 2 ] V f [ ] F max [kn] 16.2 V f 8 F max E/V Iteration [ ] E/V [MN/M 2 ] V f [ ] F max [kn] 16.2 V f 1 F max E/V Iteration [ ] Figure 3.5: Evolutionary histories of the volume fraction V f, the absorbed energy per unit volume E/V and the maximum crushing force F max for example 1 using criterion 1 (top) and criterion 2 (bottom). As the volume is decreased the energy per unit of volume appears to converge to a maximum value for both cases. Eventually the requirement for the maximum crushing force is met. For criterion 2 this means the volume starts to vibrate as the crushing force goes below and above the maximum crushing force. For criterion 1 the volume increases a bit further as the crushing force remains below the maximum crushing force. Further iterations are required to find the 12

14 optimal design for criterion 1. 4 Initial design Final Design criterion 1 Final Design criterion 2 3 Force [kn] 2 F=F* Displacement [mm] Figure 3.6: Force-displacement curves for optimal and initial full designs of example1. It is also noted that some iterations have no data points shown in figure 3.5. For criterion 1, the FEA simulations for designs 81, 92, 94, 95 and 97 did not converge. And for criterion 2, only designs 3, 33 and 6 are problematic. Because in this report elements are not actually removed but instead have reduced material properties, some designs may lead to FEA simulations that are very difficult to solve. To illustrate this problem, designs for two consecutive iterations are shown in figure 3.7. After iteration 93, some elements in the truss at the bottom are removed. This means that the element s Young s modulus and hardening modulus are reduced by a factor of 1 4. As a result these elements become very compliant and show very large deformation because of the structure s topology, making it difficult to solve the FEA simulation. The sensitivity numbers are still calculated from the last succesfull increment of the failed FEA simulation, and so the BESO program can continue. Figure 3.7: Design for criterion 1 after 93 iterations (left) and design after 94 iterations (right) The final designs for both the sensitivity for end displacement and the sensitivity for the whole displacement history are displayed in figure 3.8. Figure 3.9 shows the deformed final designs. The final design for criterion 1 is very simular to the result presented in the paper by X. Huang and Y.M. Xie [4]. The evolutionary history looks similar, and only deviates after about 5 iterations. The differences in these result can be attributed to the problems in solving some designs. There are various ways to remove elements other than reducing it s material parameters. Even the reduction factor (this study uses 1 4 ) can play a great role in the solvability of the design. Despite the differences in evolutionary history it is very likely that with further iterations, the design after 1 iterations will evolve to the final design presented in the paper by X. Huang and Y.M. Xie. The final design using the sensitivity for the whole displacement history (criterion 2 in figures 3.8 and 3.9) is identical to the results in [4]. 13

15 Figure 3.8: Final design for criterion 1 (top) and criterion 2 (bottom). Figure 3.9: Final design for criterion 1 (top) and criterion 2 (bottom) after deformation Example 2 1 mm U = 2 mm 3 mm 2 mm Figure 3.1: Design domain, dimensions and boundary conditions for example 2. The gray area denotes the initial guess design. A 2mm by 3mm beam is fixed at both ends (see figure 3.1) and a rigid object impacts at the center of the top edge. The material is elastoplastic with Young s modulus E = 1GP a, Poisson ration v =.3, yield stress σ y = 1MP a and plastic hardening modulus E p =.1E. The maximum allowable crushing force is set to 7N and the maximum crushing distance is 14

16 set to 2mm. The design domain is discretized in node plane stress elements. The evolutionary volume ratio is set to ER =.1, the admission volume ratio is AR max =.2 and the mesh-independent filter radius is R c = 3mm. In order to save computational time, an initial guess design can be set. In this example the initial design has 5% of the material in the design volume, as can be seen in figure E/V [kn/m 2 ] 15 1 V f [ ] F max [N] 5.2 V f F max E/V Iteration [ ] E/V [kn/m 2 ] 15 1 V f [ ] F max [N] 5.2 V f 2 F max E/V Iteration [ ] Figure 3.11: Evolutionary histories of the volume fraction V f, the absorbed energy per unit volume E/V and the maximum crushing force F max for example 2 using criterion 1 (top) and criterion 2 (bottom). 15

17 The evolutionary history of the optimization using both the sensitivity for end displacement (criterion 1) and the sensitivity for the whole displacement history (criterion 2) are shown in figure This example illustrates how an initial guess design can save computational time. Both cases show how the energy per unit of volume converges to a maximum value after about 4 iterations. Just like in example 1 some designs are problematic to solve using FEA (design 7 for criterion 1 an design 21 for criterion 2), but the BESO program is able to continue by using data from the solved increments. The initial guess design satisfies the force requirement (as shown in figure 3.12), and therefore the volume is initially increased. Eventually the force requirement is violated and the volume is decreased in order to satisfy the requirement again. Finally convergence is reached and the volume vibrates within 1% of the current volume as the force constraint is satisfied and violated in two successive iterations. 7 F=F* Force [N] Initial design Final Design criterion 1 Final Design criterion Displacement [mm] Figure 3.12: Force-displacement curves for optimal and initial full designs of example2. The final designs for both criteria are displayed in figure 3.13 (undeformed) and figure 3.14 (deformed). The designs and the evolutionary history are very similar to the results by by X. Huang and Y.M. Xie [4]. The initial guess design had a volume very close to the volume of the final design, and convergence is reached quite fast. This shows how an initial guess design can save computational time. Figure 3.13: Ultimate design for criterion 1 (top) and criterion 2 (bottom). 16

18 Figure 3.14: Ultimate design for criterion 1 (top) and criterion 2 (bottom) after deformation. 17

19 Chapter 4 Inverse shape optimization Inverse shape optimization aims to optimize the design or shape of an initial configurations so that it s deformed shape agrees to certain desirable criteria. Inverse shape optimization is especially useful for stent design. A stent is a mesh-like, dilatable structure which is inserted into various natural passages/conduits in the human body to counteract a disease-induced flow constriction. In order to insert a stent into the human body, it is only dilated (by means of inflation) after insertion. The shape of the deformed stent is critical to it s ability to function inside the human body. Therefore there is need for an algorithm to optimize the final shape of the stent. In this chapter a simple algorithm for inverse shape optimization using Bi-directional Evolutionary Structural Optimization (BESO) [3, 1, 6] is presented. 4.1 Objective of the optimization method In order to pose constraints on the structure s deformed shape, it is assumed the design criteria can be translated into a combination of simple desired shapes such as circular and rectangular holes in a planar design domain (such as a the shell of a thin walled cylinder). The objective of the optimization will be to gradually minimize the distance of the border of these holes to the desired border. Similar to the BESO method used in Chapter 3, the design domain is divided into M elements that can either be removed or added to the design. The solution of the optimization can be expressed as x j {, 1}, j = 1,..., M (4.1) where the design variable x j declares the presence (1) or absence () of an element. Instead of mathematically defining the objective and deriving the sensitivities, so called indicator values will be used to determine which elements should be added or removed in an optimization process that is inspired by the BESO method. Indicator number for a circular hole For a circular hole with radius R c, the indicator number is defined as α j = R c S j R c (4.2) where α j is the indicator of the j-th element and S j is the distance from the center of element j to the center of the desired circular hole (see figure 4.1). The indicator number is defined such that elements near the center of the circle will have the highest positive values. Elements close to the center of the circle that are active (x j = 1) should 18

20 .3 be the first to be removed for a stable evolutionary process. Similarly, void (x j = ) elements outside of the desired circle with the highest negative values should be the first to be added. Furthermore an element is said to be inside the circle if the element s center is located inside the circular hole. Indicator number for a rectangular hole For a rectangular hole the indicator number is defined similarly as α j = R j S j R j (4.3) where R j is now the shortest distance from the center of the desired hole to the point on the border of the rectangle that is closest to the center of the j-th element (see figure 4.1). Similar to equation (4.2), elements close the the center of the rectangle will be the first to \ \ \ \ \ \ \ R c \ \ \ \ \ R j \ \ \ Figure 4.1: Definitions of the distances used in equations (4.2) and (4.3) be deleted and elements outside of the square will have negative values proportional to their distance to the square. Figure 4.2 shows a contour-plot of the indicator numbers for a domain with a circular and a rectangular hole. y y x x Figure 4.2: Contour graph of indicator numbers for a domain with a circular hole (left) with R c = 15 and a rectangular hole (right), both with the center at (35,25). Combination of simple shapes In order to combine multiple simple holes of different size and shape, the indicator numbers of every element are calculated for each hole and the maximum value will be selected as the 19

21 .6 indicator number. α j = max(αj, 1 αj, 3..., αj N ) (4.4) where α j is the indicator number for the j-th element, and αj N is the element s indicator number for the N-th circular or rectangular hole. Figure 4.3 shows an example of combined simple holes y x Figure 4.3: Indicator numbers for a domain with a rectangular hole and a circular hole with R c = 1 on one of it s corners Evolutionary history of indicator number To further increase the stability of the optimization process, the indicator number can be averaged as α j = αk j + αk 1 j 2 (4.5) where αj k is the indicator number for the current (k-th) iteration. αk 1 j is the indicator number in the previous iteration. 4.2 Material removal/addition criterion Similar to the method discussed in chapter 3, there is no volume constraint for this problem since the shape of the undeformed design is completely unknown. A volume constraint on the deformed volume is also problematic. A stent in a blood vessel is typically dilated by an applied pressure (for example with an inflated balloon). The outer radius of the stent will become equal to the radius of the blood vessel, but the inner radius of the stent remains unknown. Instead of a volume constraint, the deformed shape is evaluated to determine if the volume needs to be increased or decreased. If the holes in the deformed shape are too large, than the volume of the design will be increased. Similarly the volume will be decreased when the holes in the deformed shape are too small. The indicator numbers are calculated for all elements (both solid and void) and then sorted according to their value. Void elements (x j = ) are switched on if α j < α th add (4.6) 2

22 And solid elements (x j = 1) are removed if α j α th del (4.7) where αdel th and αth add are threshold values for the removal and addition of elements respectively. The threshold values are determined by the following simple steps: Step 1: Let αdel th = αth add = αth. α th can be determined by V k+1. For example, if there are 4 same-size elements in the design domain and α 1 < α 2... < α 4 and V k+1 corresponds to a design with 23 elements then α th = α 23. Step 2: Calculate the admission volume ratio. If AR AR max skip step 3, otherwise recalculate αdel th and αth add as in step 3. Step 3: Calculate αadd th by first sorting the sensitivity number of void elements (x j = ). The volume of elements to be switched from to 1 will be equal to AR max multiplied by the total volume in the current design. αadd th is the sensitivity number of the element ranked just above the last added element. αdel th is then determined so that the removed volume is equal to (V k+1 V k + the volume of the added elements). In order to determine if the volume needs to be increased or decreased, the threshold indicator number of the previous iteration is recalculated in the current design (αprev.th. k ). Consequently if in the current design there are deactivated elements located outside of the desired holes (αprev.th. k < ), the volume is increased: V k+1 = V k (1 + ER) (4.8) Similarly the volume will be decreased when in the current design, activated elements are located inside the desired holes (α k prev.th. ). V k+1 = V k (1 ER) (4.9) Unlike the previous chapter, the evolutionary volume ratio (ER) is now not fixed. In order to increase stability and to prevent oscillating phenomena, the evolutionary volume ratio now depends linearly on the threshold indicator number of the previous iteration. ER k = ER init α k prev.th. (4.1) where in this chapter the initial value for the evolutionary volume ratio is chose as ER init =.5. As the optimization converges, αprev.th. k will be come closer to zero and so ER will decrease. To prevent large changes in the design that may lead to instabilities, the volume addition ratio is introduced. AR AR max (4.11) where AR is defined as the volume of added elements divided by the total volume in the current design. AR max =.2 is used in this chapter. 4.3 Implementation For the implementation of the mentioned optimization method, a combination of Abaqus with the python interface is used. The python script sets the initial design to a pre-made Abaqus model. Abaqus is then used to calculate deformation of the model. The deformed coordinates of the elements are then used to calculate the indicator numbers for every element. The volume of the next design is determined by evaluating the threshold indicator number of the previous 21

23 iteration, and the new design can be determined by evaluating the indicator numbers. The new design can now be solved using FEA and the process repeats itself until convergence of the volume is reached. Figure 3.3 shows a flow chart of the iterative process. Assign initial guess design Solve model using FEA Calculate indicator numbers (α) j Calculate the evolutionary volume ratio (ER ) k+1 No Determine volume next design (V k+1) by k evaluating α prev.th. Determine admission volume ratio (AR) Determine threshold indicator number ( α th ) Remove and add elements to assign new design Has convergence been reached? Yes Current design (V ) is optimal k Figure 4.4: Schematic of the inverse shape optimization method. Similar to chapter 3, elements are not actually removed. Instead their material properties (e.g. Young s modulus and hardening modulus) are reduced by a factor of 1 4. The python script that is used to interface with Abaqus is included in appendix A.2. 22

24 4.4 Examples Example 1 5 mm? U = 2 mm 3 mm 7 mm 5 mm Figure 4.5: Example 1: Dimensions and boundary conditions for the undeformed unknown shape (left) and the desired shape (right). A 7mm by 3mm plate is completely fixed at the center of the left side and is fixed in one direction along the rest of that side. The plate is compressed to a width of 5mm. The material is elastoplastic with Young s modulus E = 21GP a, Poisson ration v =.3, yield stress σ y = 4MP a and plastic hardening modulus E p =.2E. The design domain is discretized using node plain stress elements. The goal of the optimization method is to create a design that will deform to a shape with a circular hole with a diameter of D = 3mm in it s center. The initial design has 1% of the material ER [%] 6 4 V f [ ].6.4 V f k α prev.th. ER.6.4 [ ] α prev.th. k Iteration [ ] Figure 4.6: Example 1: Volume fraction, threshold value for the indicator number and evolutionary volume ratio. 23

25 Figure 4.7: Example 1: Undeformed (left) and deformed shape (right) of the final design. Figure 4.6 shows the evolutionary history of the optimization process. Initially αprev.th. k = 1 and therefore the volume is decreased in order to increase the size of the hole. As αprev.th. k approaches, the evolutionary volume ratio (ER) decreases as well. As a result the volume decreases more slowly until αprev.th. k and the evolutionary volume ratio becomes so small that no more elements can be added or removed. The final shape after 5 iterations is presented in figure 4.7. The initial elliptical hole deforms under compression to an almost circular hole. The design can be further improved by decreasing the element-size. This result shows that the method is very successful for the inverse shape optimization of circular holes under compression Example 2 Example 2A 5 mm? U = 2 mm 35 mm 35 mm 7 mm 5 mm Figure 4.8: Example 2A: Dimensions and boundary conditions for the undeformed unknown shape (left) and the desired shape (right). A 7mm by 3mm plate is completely fixed at the center of the left side and is fixed in one direction along the rest of that side. The plate is compressed to a width of 5mm. The material is elastoplastic with Young s modulus E = 21GP a, Poisson ration v =.3, yield stress σ y = 4MP a and plastic hardening modulus E p =.2E. The design domain is discretized using node plain stress elements. The goal of the optimization method is to create a design that will deform to a shape with a rectangular hole with a dimensions: 35mm by 35mm in it s center. The initial design has 1% of the material. 24

26 1 1.8 V f 8.8 k α prev.th. ER.6 ER [%] 6 4 V f [ ] [ ] k α prev.th Iteration [ ] Figure 4.9: Example 2A: Volume fraction, threshold value for the indicator number and evolutionary volume ratio. The deformed designs of iteration 94 and 95 are displayed to illustrate the oscillating behavior. The evolutionary history in figure 4.9 shows oscillations in the threshold value for the indicator number. This indicates that the size of the hole in the deformed material oscillates. The volume fraction does not show this behavior. The volume keeps decreasing because α k prev.th. remains larger than zero. Therefore it must be concluded that the oscillatory behavior is caused by the way the geometry deforms. Figure 4.1: Example 2A: Deformed shape of the design for two consecutive iterations. 94 iterations (left) and 95 iterations (right). Figure 4.1 shows the deformed shape of two consecutive iterations. After 94 iterations the deformed shape is very close to the desired shape, and the top and bottom edges buckle away from the center. Because there are some elements inside the desired hole at iteration 94, most of them are removed for the next iteration. As a result of this removal the new design now buckles in opposite direction. Because of the buckling in opposite direction, the deformed shape 25

27 of design 95 is now more undesirable than design 94 was. Again the elements inside the desired shape that are closest to the center will be removed first, and a shape similar to design 94 will emerge. This periodic occurrence is clearly visible in figure 4.9. Figure 4.11: Example 2A: Undeformed (left) and deformed shape (right) of the design after 1 iterations. Figure 4.11 shows the final design after 5 iterations. The deformed shape shows buckling both towards the center and away from the center. This is caused by a slight a-symmetry, and can be prevented by optimizing only a quarter of the design. This example illustrates how buckling is a severe limitation for the inverse shape optimization of rectangular holes under compression. The magnitude of the problem will depend on the size of the hole in relation to the part and the amount of compression. For example the problem will not exist if the same plate will be used for the inverse shape optimization of a very small rectangular hole. Example 2B 5 mm? U = 2 mm 2 mm 2 mm 7 mm 5 mm Figure 4.12: Example 2B: Dimensions and boundary conditions for the undeformed unknown shape (left) and the desired shape (right). This example will demonstrate how buckling does not cause problems with inverse shape optimization of relatively small rectangular holes under compression. Similar to example 2A, a 7mm by 3mm is completely fixed at the center of the left side and is fixed in one direction along the rest of that side. The loading conditions and material properties are as before. The goal of the optimization method is to create a design that will deform to a shape with a rectangular hole with a dimensions: 2mm by 2mm in it s center. The initial design has 1% of the material. 26

28 ER [%] 6 4 V f [ ].6.4 V f k α prev.th. ER.4.2 [ ] k α prev.th Iteration [ ] Figure 4.13: Example 2B: Volume fraction, threshold value for the indicator number and evolutionary volume ratio. The evolutionary history is displayed in figure Materials is removed and after 5 iterations αprev.th. k. Buckling only takes place away from the center, and the final design in figure 4.14 has an almost perfect rectangular hole. Figure 4.14: Example 2B: Undeformed (left) and deformed shape (right) of the design after 5 iterations. 27

29 4.4.3 Example 3 1 mm? U = 4 mm 4 mm 4 mm 8 mm Figure 4.15: Example 3: Dimensions and boundary conditions for the undeformed unknown shape (left) and the desired shape (right). A 4mm by 1mm plate is completely fixed at the center of the left side and is fixed in one direction along the rest of that side. Tension is applied on the right side of the plate to increase it s width to 8mm. The material is elastoplastic with Young s modulus E = 21GP a, Poisson ration v =.3, yield stress σ y = 4MP a and plastic hardening modulus E p =.2E. The design domain is discretized using node plain stress elements. The goal of the optimization process is to create a design that under tension will deform to a shape with a circular hole of diameter of D = 4mm in it s center. The initial design will be a projection of the desired shape on the elements, assuming they all deform uniformly. 28

30 ER [%] 6 4 V f [ ] [ ] α prev.th. k 2.2 V f k α prev.th..6 ER Iteration [ ] Figure 4.16: Example 3: Volume fraction, threshold value for the indicator number and evolutionary volume ratio. The designs at some iterations are also displayed to illustrate the shape evolution. The evolutionary history (4.16) shows that as αprev.th. k initially approaches zero very rapidly, the value then begins to change periodically. During this periodic occurrence the volume fraction changes only slightly as the evolutionary volume ratio is very small. As a result only a few elements are removed or added in these iteration steps. The periodic occurrence is caused by elements that in one iteration become candidates for removal because they are inside the desired hole, while on the next iteration the deformation is slightly different and the elements are now outside the desired hole and again candidates to be added back to the design. The initial design has a volume fraction of V f =.79, but does not appear to be a good guess for the final design. Initially the hole is too large as αprev.th. k <, and therefore material is added. The final design after 5 iterations (figure 4.17) has a volume fraction of V f 1 and is basically a gap the length of diameter of the desired circle. 29

31 Figure 4.17: Example 3: Undeformed (left) and deformed shape (right) of the design after 1 iterations. Because the desired shape has a circular hole, there will be stress concentrations above and below the hole. Therefore the elements in that area will have a lot of deformation (see figure 4.17) while other element show little deformation. Since the removed elements are very compliant, they show extreme deformation. As a result the solution simply is a gap in the plate, creating a hole that is close to a circular shape. Optimization with a full initial design yields similar results. 4 mm? U = 4 mm Example 4 1 mm mm 4 mm 8 mm Figure 4.18: Example 4: Dimensions and boundary conditions for the undeformed unknown shape (left) and the desired shape (right). A 4mm by 1mm plate is completely fixed at the center of the left side and is fixed in one direction along the rest of that side. Tension is applied on the right side of the plate to increase it s width to 8mm. The material is elastoplastic with Young s modulus E = 21GP a, Poisson ration v =.3, yield stress σy = 4M P a and plastic hardening modulus Ep =.2E. The design domain is discretized using node plain stress elements. 3

32 The goal of the optimization process is to create a design that under tension will deform to a shape with a rectangular hole with dimensions: 4mm by 4mm in it s center. The optimization is done for an initial full design and an initial guess design. The initial guess design will be a projection of the desired shape on the elements, assuming they all deform uniformly V f ER [%] 6 4 V f [ ].6.4 k α prev.th. ER.3 [ ] k α prev.th Iteration [ ] Figure 4.19: Example 4 with full initial design: Volume fraction, threshold value for the indicator number and evolutionary volume ratio. Figure 4.2: Example 4 with full initial design: Undeformed (left) and deformed shape (right) of the design after 1 iterations. 31

33 The evolutionary history for the full design shows that the design initially decreases in volume and eventually increases to about 97% of the design domain. The final design in figure 4.2 deforms to have a rectangular hole of the desired dimensions. Similar to the previous example the design has a gap the width of one element. Because a rectangular hole has no stress concentrations above and below the hole, the deformation does not localize V f ER [%] 6 4 V f [ ].6.4 k α prev.th. ER.3 [ ] k α prev.th Iteration [ ] Figure 4.21: Example 4 with initial guess design: Volume fraction, threshold value for the indicator number and evolutionary volume ratio. 32

34 Figure 4.22: Example 4 with initial guess design: Undeformed (left) and deformed shape (right) of the design after 1 iterations. The final result for optimization with the initial guess design is very similar. However because of the difference in evolutionary history, the gap in the final design now has a width of two elements. The initial guess design is not very close to the final design, and therefore even more iterations are needed for convergence that with the initial full design. It is noted that the nodes to the left and right of the rectangular hole in both solutions show little deformation. Therefore if the goal would have been a rectangular hole with a smaller width, a satisfactory solution may not have been found (if it even exists at all). 33

35 Chapter 5 Conclusion In this report a new shape optimization method (called inverse shape optimization) is introduced that can be used to optimize the geometry of stents. The method is based on the bi-directional evolutionary structural (BESO) optimization method. The implementation of the BESO method was benchmarked by reproducing the results from a paper by Huang, Xie and Lu [4] on topology optimization of energy-absorbing structures. The topology optimization examples discussed in chapter 3 lead to results that are almost identical as the paper by Huang et. al.. In this report the BESO method is implemented in such a way that elements are not actually removed, but instead their Young s and hardening modulus are reduced by a factor 1 4. Removed elements become extremely compliant, and this becomes problematic when some designs have extreme deformation of these removed elements. These designs are very difficult to solve using FEA. But despite some failed FEA simulations, the BESO method is still able to continue and optimal designs are found. The inverse shape optimization method aims to optimize the undeformed shape of a product such that it s deformed shape agrees to certain criteria. In order to pose constraints on the structure s deformed shape, it is assumed the design criteria can be translated into a combination of simple desired shapes such as circular and rectangular holes in a planar design domain (such as a the shell of a thin walled cylinder). The objective of the optimization is to gradually minimize the distance of the border of these holes to the desired border. So called indicator numbers are calculated for elements in the design domain to determine what elements should be removed or added. Elements close to the center of a desired hole will be the first be removed, and elements outside the desired hole that are the farthest away from the center will be the first to be added back to the design. The examples in chapter 4.4 demonstrate the inverse shape optimization method under large deformation in both compression and tension. The examples show that the method is very successful for the inverse shape optimization of circular holes under compression. Inverse shape optimization of rectangular holes under compression is limited because of the deformation mechanism. If the size of the rectangular hole in relation to the entire part is too large, the method becomes unstable because the direction of buckling may change with very small changes to the design. This does not occur when the size of the rectangular hole is sufficiently small with respect to the entire part. Inverse shape optimization of a circular hole under tension is problematic. The circular shape of the desired hole causes stress concentrations. Elements near the stress concentrations will have a lot of deformation, and as a result the final design of the optimization method is a gap the with of one element. After deformation this gap does not deform to a perfect circle. 34