Biomimetic Approach to Compliance Optimization and Multiple Load Cases
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Abstract
The variational approach to shape optimization in linearized elasticity is used in order to improve convergence of a known heuristic algorithm. The speed method of shape optimization is applied to obtain necessary optimality conditions for representative test examples. The algorithm originates from the biomimetic approach to compliance optimization. The trabecular bone adapts its form to mechanical loads and is able to form structures that are lightweight and very stiff at the same time. In this sense, it is a problem pertaining to both the nature or living entities which is similar to structural optimization, especially topology optimization. The paper presents the biomimetic approach, based on the trabecular bone remodeling phenomenon, with the aim of minimizing the compliance in multiple load cases. The method employed aims at minimizing the energy and combines structural evolution inspired by trabecular bone remodeling and the shape gradient framework, with strict analysis based on functionals in the 3dimensional elasticity model. The method is enhanced to handle the problem of structural optimization under multiple loads. The new biomimetic approach does not require volume constraints. Instead of imposing volume constraints, shapes are parameterized by the assumed strain energy density on the structural surface. The stiffest design is obtained by adding or removing material on the structural surface in virtual space. Structural evolution is based on shape gradient approximation by the speed method, and it is separated from the finite element method of the model solution. Numerical examples confirm that the heuristic algorithm for structural optimization is efficient.
Keywords
Biomimetic structural optimization Shape derivative Multiple loadsMathematics Subject Classification
35C20 35J15 35S05 49J40 49Q121 Introduction
The variational approach to shape and topology optimization can be developed, e.g., within the speed and the topological derivative methods. In particular, the topological derivative method is used to this end [1, 2, 3]. Our aim in this paper is to improve the convergence of the biomimetic approach to compliance optimization and multiple load cases by an application of the speed method. This means that we introduce a variational problem for which the proposed optimality criterion of biomimetic approach becomes a necessary optimality condition. Such an approach is known in the theory of free boundary problems.
The field of structural optimization is still a relatively new area but with many applications in the automotive, aerospace, civil engineering, machine design and other engineering fields. The classical approach to structural optimization can be found, e.g., in the monographs [4, 5]. Many researchers must face challenges related to structural optimization. One of the crucial ones is to develop structural optimization methods that are appropriate for multiple load issues.
From an engineer’s point of view, single load case problems are rather rare. A more common problem, but also more challenging in terms of mechanical design, is the one of structural optimization under multiple loads. The biomimetic optimization method should be useful as, in real life, multiple independent loads are always present.
It is widely believed that the trabecular bone adapts its form to mechanical loads following Wolff’s law [6, 7]. A healthy tissue of the trabecular bone has a very sophisticated structure. The tissue forms a network of beams called trabeculae, and this structure is capable of handling a wide range of loads. The length of a trabecula can be 100 or 200 \(\upmu \hbox {m}\), whereas its diameter is approximately 50 \(\upmu \hbox {m}\). This structure is continually rebuilding itself so that the whole bone tissue is replaced within a few years. The process is called trabecular bone adaptation or remodeling.
Wolff’s law is not a mechanical law as such but rather the hypothesis that the trabecular bone is a selfoptimizing material. In addition, other researchers suggested [8, 9] that the trabecular tissue can adapt its structural form in reaction to external loads. In this sense, it is a problem which resembles structural optimization, especially topology optimization.
The strain energy density on the structural surface also appears in the area of optimization research, which is distant from biomechanical studies [10, 11]. The most important theorem concerning the energy density on the structural surface states that for the stiffest design, the energy density must be constant along the designed shape. The new formulation of biomimetic optimization based on the trabecular bone phenomenon used in [12] was established with the use of the shape derivative concept. The mathematical analysis of the socalled speed method in shape optimization is performed, e.g., in [13].
2 Structural Design with Multiple Load Cases
A multiple load problem arises when a reallife design has to be developed. In topology optimization, a common procedure uses minimization of a weighted sum of different loads. The compliance for each load case is computed, and then, the multiload problem is replaced by minimization of the weighted sum of the compliances under each load, as first presented by Sigmund [14]. For optimization of the worst possible case, another approach is needed. One of the possible methods is the bound formulation approach [15, 16, 17]. The main feature of the approach is the introduction of an upper bound for other objectives and minimizing the upper bound. Another approach uses the aggregation technique [18, 19]. The main advantage of the constraint aggregation method is that all load cases contribute throughout the optimization process, owing to which the process does not lead to local minima.
Relying on the approach presented in [12], using biomechanical models of the trabecular bone remodeling directly as a base for structural optimization, we will discuss a design with multiple load cases. In general, there is no solution to the problem of stiffest design (compliance minimization). If the volume of an object is increasing, the compliance is decreasing. Hence, in the standard approach to energybased topology optimization, an additional constraint has to be added. Usually, the volume of material is limited. When using volume constraints, additional stress constraints should be introduced. As a result, topology optimization has to be separated from shape and size optimization within the numerical procedure [17]. In the case of the biomimetic approach, the role of an additional constraint is played by the strain energy density on the structural surface, and the volume or structural mass results from the optimization procedure. In the standard SIMP topology optimization method, the volume constraint is used. Also, in the stressbased topology optimization approach [20, 21], the volume constraint is present. To start the optimization procedure, the volume constraint has to be imposed. In the biomimetic approach, instead of imposing a volume constraint, we parameterize shapes by the assumed energy density, which may be quite accurately predicted from yield criteria.
3 Problem Setting
4 Insensitivity Zone Concept
The biomimetic approach based on trabecular bone remodeling allows comprising optimization of size, shape and topology in one numerical procedure, where the lazy zone concept is an important component of the algorithm. The lazy zone concept was originally proposed by Carter [22], as an enhancement of the mechanostat theory of Frost [23]. The concept was also exploited by other researchers [9, 24]. The stiffest design is obtained by adding or removing material on the structural surface in virtual space, mimicking the natural process of trabecular bone remodeling [25]. Bone mass increases above a certain level of mechanical stimulation (measured by the strain energy density on the tissue surface) and decreases below a certain energy level. When the strain energy density on the structural surface is between these two levels, the bone mass is maintained, and this rate of bone mechanical stimulation is called adaptation or lazy zone because then the bone does not react to changes in the mechanical stimulation level. The processes of resorption and formation are coupled in basic multicellular units, regions smaller than the single trabecula where a sequence of successive bone tissue resorption and formation occurs, and this is a local change. However, bone formation depends on global mechanical stimulation of the whole bone [8, 9].
5 Heuristic Algorithm
The numerical results of [26] show a similarity between trabecular bone remodeling and structural optimization. The results have no biological background, but have been rigorously proven mechanically and mathematically. Nonetheless, trabecular bone remodeling models could be used to realize the optimization procedure.

it is assumed that the energy density \(\varvec{\sigma }({\mathbf {u}}): \varvec{\varepsilon }({\mathbf {u}})\) has a constant value \(\lambda \) on \(\varGamma _v\);

if at a given point on \(\varGamma _v\) this density is greater than \(\lambda +s\), then the boundary is moved outside;

if at a given point on \(\varGamma _v\) this density is smaller than \(\lambda s\), then the boundary is moved inside;

these steps are repeated until equilibrium is achieved;

the value of \(\lambda \) is modified if the final design is unsatisfactory.

If \(\kappa > 0\) and \(F > {\bar{F}}\), then after a biomimetic modification the boundary is additionally moved inside \(\varOmega \) by 50% of the biomimetic step.

If \(\kappa < 0\) and \(F > {\bar{F}}\), then after a biomimetic modification the boundary is additionally moved outside \(\varOmega \) by 50% of the biomimetic step.
6 MultipleObjective Problems
The systematic approach proposed in this article consists in formulating two versions of scalarized multipleobjective problems.
The problems will be transformed suitably and analyzed using the speed method.
7 Definition of the Speed Method
8 Problem Transformation
Problem 1 may be transformed into an equivalent form:
9 Shape Modification Using Shape Derivative Without Volume Constraint

it is assumed that the energy density \(\varvec{\sigma }({\mathbf {u}}): \varvec{\varepsilon }({\mathbf {u}})\) has a constant value \(\lambda _0\) on \(\varGamma _v\);

the energy densities \(\varvec{\sigma }({\mathbf {u}}_1): \varvec{\varepsilon }({\mathbf {u}}_1)\) and \(\varvec{\sigma }({\mathbf {u}}_2): \varvec{\varepsilon }({\mathbf {u}}_2)\) are computed for loads \({\mathbf {t}}_1\) and \({\mathbf {t}}_2\) acting on the same boundary \(\varGamma _1\);

the average energy density is \(\frac{1}{2} \varvec{\sigma }({\mathbf {u}}_1):\varvec{\varepsilon }({\mathbf {u}}_1) + \frac{1}{2} \varvec{\sigma }({\mathbf {u}}_2):\varvec{\varepsilon }({\mathbf {u}}_2) \) for loads \({\mathbf {t}}_1\) and \({\mathbf {t}}_2\);

if at a given point on \(\varGamma _v\) this density is greater than \(\lambda _0+s\), the boundary is moved outside;

if at a given point on \(\varGamma _v\) this density is smaller than \(\lambda _0s\), the boundary is moved inside;

these steps are repeated until equilibrium is achieved;

the value of \(\lambda _0\) is modified if the final design is unsatisfactory.

If \(\kappa > 0\) and \(F > {\bar{F}}\), then after a biomimetic modification the boundary is additionally moved inside \(\varOmega \) by 50% of the biomimetic step.

If \(\kappa < 0\) and \(F > {\bar{F}}\), then after a biomimetic modification the boundary is additionally moved outside \(\varOmega \) by 50% of the biomimetic step.
For Problem 2 the algorithm will be the same, with \(\frac{1}{2} \varvec{\sigma }({\mathbf {u}}_1):\varvec{\varepsilon }({\mathbf {u}}_1) + \frac{1}{2} \varvec{\sigma }({\mathbf {u}}_2):\varvec{\varepsilon }({\mathbf {u}}_2) \) replaced by \(\alpha _1 \varvec{\sigma }({\mathbf {u}}_1):\varvec{\varepsilon }({\mathbf {u}}_1) + \alpha _2 \varvec{\sigma }({\mathbf {u}}_2):\varvec{\varepsilon }({\mathbf {u}}_2)\).
10 Numerical Examples
To compute the strain energy density on the structural surface, the finite element method is used. (The details of the numerical environment can be found in [12].) For the multiple load case simulation, the problem with a square design domain with a central square rigid support was chosen. The reason was the existence of an exact analytical solution examined in many papers [27, 28, 29].
Two different load cases were examined, depicted in Fig. 1. The forces in each case are of the same magnitude. Material parameters were taken as follows: Young’s modulus \(2\times 10^{11}\) Pa, Poisson’s ratio 0.3 and the size of the insensitivity zone 80–180 MPa. Since the biomimetic method works in 3D only, the design domain is in fact 3dimensional, but due to very small thickness it can be treated as quasi 2dimensional.
The results (selected steps) of the optimization procedure for the multiple load cases for force sets P and Q are presented in Fig. 2. The optimal configuration obtained is similar to the results presented previously by other researchers.
To compare our approach with other results derived by a more classical approach, a widely investigated Tbracket structure was chosen. The Tbracket problem model is presented in Fig. 3. Two load cases were analyzed. The forces for both load cases were of the same magnitude, and the material parameters were taken as in the previous example. For this example, 82 iterations were needed. The structural evolution was based on shape gradient approximation by the speed method, and it was separated from the finite element method computations. Each iteration included generation of a mesh (tetrahedral elements) and finite element calculations for each load case. The computational time for the initial configuration of 93,120 elements was 10 s (wallclock time) for the mesh generation and 25 s for the finite element calculations for each load case. For the final configuration, there were 42,396 elements, 5 s for meshing and 20 s for structural calculation. The calculations were carried out with the use of our own mesh generation tool Cosmoprojector and the parallel finite element solver based on Message Passing Interface (MPI). The computations were run on 4 CPUs with the use of the cluster (16 GB RAM per node). The details of the mesh generation method can be found in [12].
In order to test the procedure for 3D problems, two different load cases were examined, and the design domains are presented in Fig. 5. It was assumed that all nodes of the back wall of the domain box could be fixed (clamped wall), and the bending force was applied to the middle of the opposite, front wall. The first load case was with a vertical bending force, and the second with the same definition of boundary conditions and a horizontal bending force. All loads were of the same magnitude. Material parameters and the size of the insensitivity zone were assumed to be as in the previous example.
The biomimetic optimization system, as in the case of living creatures, is 3dimensional, and the definition of the design domain is not needed. Moreover, the definition of boundary conditions could be different for each step of the simulation. Instead of starting the computations from a box domain, it is possible to start from just a stick, connecting the clamped wall (the potential surface, where no displacement boundary conditions could be defined) as a starting configuration. The results under such assumptions are presented in Fig. 7.
11 Conclusions
In this paper, a new formulation of biomimetic optimization based on trabecular bone remodeling applied to a multiple load problem is presented. The stiffest design is obtained by adding or removing material on the structural surface in virtual space. Three elements complete the formula driving the process of adding or removing material: nonlocal influence of boundary modification, spatial variability of the function penalizing deviation from the assumed strain energy density on the structural surface and measuring the increase or decrease in the area of the surface itself, by measuring the surface curvature. The structural evolution is based on shape gradient approximation by the speed method, and it is separated from finite element method computations. The finite element method is used to compute the distribution of the strain energy density only. An important advantage of our approach is that optimization results are in the form of a functional configuration. Accordingly, the assumed value of the strain energy density on the part of the boundary subject to modification can be related to material properties. Instead of imposing a volume constraint, we parameterize shapes by energy density, which may be quite accurately predicted from yield criteria. The functional configurations during the process of optimization allow including size, shape and topology optimization in one numerical procedure.
The 2D result for the common benchmark is in line with previous results by other researchers [27, 28, 29]. The 3D result shows the configuration for a real material. In view of the above, our 3D results with two perpendicular bending forces as multiple loads could be a good benchmark for testing other approaches to multiple load optimization.
Notes
Acknowledgements
This work was supported by the Polish National Centre for Research and Development under Grant No. TANGO 1/266483/NCBR/2015 and by the computational Grant No. 362 (Biomimetic approach to compliance optimization and multiple load cases) of Poznań Supercomputing and Networking Center (PCSS), Poznań, Poland. J.S. acknowledges the partial support of the French Agence Nationale de la Recherche (ANR), under Grant ANR17CE080039 (Project ArchiMatHOS).
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