On topology optimization of design-dependent pressure-loaded three-dimensional structures and compliant mechanisms
OOn topology optimization of design-dependentpressure-loaded 3D structures and compliant mechanisms
Prabhat Kumar ∗ , and Matthijs Langelaar † ∗ Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, 2800Kgs. Lyngby, Denmark † Department of Precision and Microsystems Engineering, Delft University of Technology, 2628CDDelft, The Netherlands
Abstract:
This paper presents a density-based topology optimization method for design-ing 3D compliant mechanisms and loadbearing structures with design-dependent pressureloading. Instead of interface-tracking techniques, the Darcy law in conjunction with adrainage term is employed to obtain pressure field as a function of the design vector. Toensure continuous transition of pressure loads as the design evolves, the flow coefficientof a finite element is defined using a smooth Heaviside function. The obtained pressurefield is converted into consistent nodal loads using a transformation matrix. The pre-sented approach employs the standard finite element formulation and also, allows consistentand computationally inexpensive calculation of load sensitivities using the adjoint-variablemethod. For compliant mechanism design, a multi-criteria objective is minimized, whereasminimization of compliance is performed for designing loadbearing structures. Efficacyand robustness of the presented approach is demonstrated by designing various pressure-actuated 3D compliant mechanisms and structures.
Keywords:
Topology Optimization; Three-dimensional Compliant Mechanisms; Design-dependent Pressure Loading; Darcy Law; Three-dimensional Structures
Nowadays, the use of topology optimization (TO) approaches in a wide variety of designproblems for different applications involving single and/or multi-physics is continuouslygrowing because of their proven capability and efficacy [1]. These methods determine anoptimized material layout for a given design problem by extremizing the desired objective(s) Corresponding author: [email protected], [email protected] a r X i v : . [ c s . C E ] S e p nder a given set of constraints. Based on the considered loading behavior, they can beclassified into approaches involving design-independent (invariant) loads and methods con-sidering design-dependent forces. The latter situation often arises in case of aerodynamicloads, hydrodynamic loads and/or hydrostatic pressure loads, in various applications in-cluding aircraft, pumps, ships and pneumatically actuated soft robotics [2, 3, 4]. ManyTO methods exist for the former loading scenarios, whereas only few methods consider-ing design-dependent loading behaviors have been reported [5]. Design-dependent loadsalter their location, direction and/or magnitude as optimization progresses and thus, poseunique challenges [6]. Those challenges get even more pronounced in a 3D TO setting [7, 8].Here, our motive is to present an efficient and robust TO method suitable for 3D designproblems including loadbearing structures and small deformation compliant mechanismsinvolving design-dependent pressure loads.Compliant mechanisms (CMs), monolithic structures incorporating flexible regions, rely ontheir elastic deformation to achieve their mechanical tasks in response to external stimuli.These mechanisms furnish many advantages over their rigid-body counterparts [9, 10, 11].Since they are monolithic designs, they require lower assembly and manufacturing cost andby comprising fewer parts and interfaces, they have comparatively less frictional, wear andtear losses. However, designing CMs is challenging, particularly in case of design-dependentloading. Therefore, dedicated TO approaches are desired.To design a CM using TO, in general, an objective stemming from a flexibility measure (e.g.,output/desired deformation) and a load bearing characteristic (e.g., strain-energy, stiffness,input displacement constraints and/or stress constraints) is optimized [12]. The associateddesign domain is described using finite elements (FEs), and in a typical density-basedTO method, each FE is associated with a design variable ρ ∈ [0 , ρ = 1 indicates solid phase of an FE, whereas its void state is represented via ρ = 0. Various applications of such mechanisms designed via TO in the case of design-independent loads can be found in [13, 14, 15, 16, 17] and references therein. However,in case of design-dependent loading different approaches are required. Figure 1 illustratesschematic design problems for a pressure-actuated CM and a pressure-loaded structure.For clarity of presentation, these are shown in 2D. A key characteristic of the problemsis that the loaded surface is not predefined, but subject to change during the TO designprocess. Accurate calculation of load sensitivities is therefore important for these problems.Hammer and Olhoff [6] were first to conceptualize design-dependent pressure-loaded For the sake of simplicity, we write only pressure load(s) instead of design-dependent pressure load(s) ,henceforward, in this paper . p Γ p Compliant Mechanism Output Ω Pressure loadsFixed Fixed Γ p b (a) Ω Pressure loadsFixed Loadbearing structure Γ p Γ p Γ p b Fixed (b)
Figure 1: Schematic representation of design domains Ω for finding a pressure-actuated(depicted by gray dash-dotted arrows) optimized compliant mechanism (black solid con-tinuum) and pressure-loaded structure (black solid continuum) in 2D. Γ p and Γ p bound-aries (surfaces) indicate surfaces with zero and pressure loads. Γ p b is the curve where thepressure loads are applied in the optimized designspressure-loaded boundary explicitly, in addition to the design variables, and they also opti-mized pressure load variables during optimization. For an overview of 2D pressure-loadedTO approaches for designing structures and/or CMs, we refer to our recent paper on thissubject [5].Locating well-defined surfaces for applying pressure loads, relating pressure loads to thedesign vector and evaluating consistent nodal forces and their sensitivities with respect tothe design vector are the central issues when considering design-dependent pressure loadsin a TO setting. Compared with 2D design problems, providing a suitable solution tothese challenges becomes even more complicated and involved for 3D design problems. Inaddition, difficulties associated with designing CMs using TO [9] contribute further to theabove-mentioned challenges. Only few approaches are available in the literature for 3DTO problems involving pressure loads [7, 8, 20, 21, 22]. Du and Olhoff [7] divided a 3Ddomain into a set of parallel 2D sections using a group of parallel planes to locate validloading curves using their earlier 2D method[18]. Thereafter, they combined all these validloading curves to determine the appropriate surface to construct the pressure loads for the3D problem. A finite difference method was employed for the load sensitivities calcula-tion, which is computationally expensive. The boundary identification scheme presentedby Zhang et al. [8] is based on an a priori selected density threshold value. Similar to the3pproach by Du and Olhoff[7], a 3D problem is first transformed into series of 2D problemsusing a group of parallel planes to determine valid loading surface. The loading surfaceis constructed by using the facets of FEs, and the load sensitivities are not accounted forin the approach. Steps employed in Refs. [7, 8] for determining pressure loading surfacesmay not be efficient and economical specially for the large-scale 3D design problems. Yanget al. [20] used the ESO/BESO method in their approach. Sigmund and Clausen [21]employed a displacement-pressure based mixed-finite element method and the three-phasematerial definition (solid, void, fluid) in their approach. They demonstrated their methodby optimizing pressure-loaded 2D and 3D structures. FEs used in the mixed-finite elementmethods have to fulfill an additional Babuska-Brezzi condition for stability [23]. Pangani-ban et al. [22] proposed an approach using a non-trivial FE formulation in associationwith a three-phase material definition. They demonstrated their approach by designing apressure-actuated 3D CM in addition to designing pressure-loaded 3D structures.In order to combine effectiveness in 3D CM designs under pressure loads, ease of implemen-tation and accuracy of load sensitivities, we herein extend the method presented by Kumaret al. [5] to 3D design problems involving both structures and mechanisms. With this, weconfirm the expectation expressed in our earlier study, that the method can be naturallyextended to 3D. The approach employs Darcy’s law with a drainage term to identify loadingsurfaces (boundaries) and relates the applied pressure loads with the design vector ρ . Thedesign approach solves one additional PDE for pressure field calculation using the standardFE method. Because this involves a scalar pressure field, the computational cost is consid-erably lower than that of the structural analysis. The pressure field is further transformedto consistent nodal loads by considering the force originating due to pressure differences asa body force. Thus pressure forces are projected over onto a volume rather than a bound-ary surface, but due to the Saint-Venant’s principle this difference is not relevant whenevaluating global structural performance. Note that this force projection is conceptuallyaligned with the diffuse boundary representation commonly applied in density-based TOmethods. The load sensitivities are evaluated using the adjoint-variable method. For de-signing loadbearing structures, compliance is minimized, whereas a multi-criteria objective[12] is minimized for CMs.The layout of the paper introduce 3D method as follows. Sec. 2 presents the proposedpressure loading formulation in a 3D setting and the transformation of pressure field toconsistent nodal loads. A 3D test problem is also discussed for indicating the role ofthe drainage term in the presented approach as well as the influence of other problemparameters. The considered topology optimization problem definitions with the associatedsensitivity analysis are introduced in Sec. 3. Sec. 4 subsequently presents several designproblems in 3D settings, including loadbearing structures and compliant mechanisms andtheir optimized continua. Lastly, conclusions are drawn in Sec. 5.4 Modeling of Design-dependent pressure loads
In a TO setting, to determine the optimized design of a given problem, the material layoutof the associated design domain Ω evolves with the optimization iterations. Consequently,in the beginning of the optimization with design-dependent loads, it may be difficult tolocate a valid loading surface where such forces can be applied. In this section, we presenta 3D FE modeling approach to determine a pressure field as a function of the design vector ρ using the Darcy law, which allows locating the loading surfaces implicitly. Evaluation ofthe consistent nodal loads from the obtained pressure field is also described. In this subsection, first the Darcy-based pressure projection formulation is summarized,following our earlier 2D paper[5]. The Darcy law which determines a pressure field througha porous medium is employed. The fluidic Darcy flux q in terms of the pressure gradient ∇ p , the permeability κ of the medium, and the fluid viscosity µ can be written as q = − κµ ∇ p = − K ∇ p, (1)where K is called the flow coefficient which defines the ability of a porous medium to permitfluid flow. In a density-based TO setting, each FE is characterized by a density variablethat interpolates its material properties between those of the solid or void phase. Thenit is natural to represent the flow coefficient of an FE with index i in terms of its filtered(physical) material density ˜ ρ i [24] and the flow coefficients of its void and solid phases suchthat it has a smooth variation within the design domain. Herein, we define K ( ˜ ρ i ) as K ( ˜ ρ i ) = K v (1 − (1 − (cid:15) ) H κ ( ˜ ρ i , η κ , β κ )) , (2)where H κ ( ˜ ρ i , η κ , β κ ) = (cid:18) tanh ( β κ η κ ) + tanh ( β κ ( ˜ ρ i − η κ ))tanh ( β κ η κ ) + tanh ( β κ (1 − η κ )) (cid:19) , (3)is the smooth Heaviside function. η κ and β κ are parameters which control the position ofthe step and the slope of K ( ˜ ρ i ), respectively. Further, K s K v = (cid:15) is termed flow contrast whichis set to 10 − as motivated in Appendix 5, where K v and K s represent flow coefficients forvoid and solid elements, respectively.As topology optimization progresses, it is expected that the pressure gradient should getconfined within the solid FEs directly exposed to the pressure loading. This cannot beachieved using Eq. 1 only (see Sec. 2.3), as it tends to distribute the pressure drop through-out the domain. Therefore, we conceptualize a volumetric drainage quantity Q drain tosmoothly drain out the pressure (fluid) from the solid FEs downstream of the exposed5urface. It is defined in terms of the drainage coefficient D ( ˜ ρ i ), the pressure field p , andthe external pressure p ext as Q drain = − D ( ˜ ρ i )( p − p ext ) . (4)The drainage coefficient D ( ˜ ρ i ) is determined using a smooth Heaviside function such thatpressure drops to zero for an FE with ˜ ρ e = 1 as D ( ˜ ρ i ) = d s H d ( ˜ ρ i , η d , β d ) , (5)where β d and η d are the adjustable parameters, and H d ( ˜ ρ i , η d , β d ) is defined analogous toEq. 3. The drainage coefficient of a solid FE, d s , is used to control the thickness of thepressure-penetration layer and is related to K s as [5]d s = (cid:18) ln r ∆ s (cid:19) K s , (6)where r is the ratio of input pressure at depth ∆s, i.e., p | ∆ s = rp in . Further, ∆ s , thepenetration depth for the pressure field, can be set equal to the width or height of few FEs.This additional drainage term ensures controlled localization of the pressure drop at theexposed structural boundary. This section presents the 3D FE formulation for the pressure field and corresponding con-sistent nodal loads. The basic state equilibrium equation for the incompressible fluid flowwith a drainage term can be written as (Fig. 2) ( q x dydz + q y dzdx + q z dxdy + Q drain dV ) = (cid:18) q x dydz + q y dzdx + q z dxdy + (cid:18) ∂q x ∂x + ∂q y ∂y + ∂q z ∂z (cid:19) dV (cid:19) , or, ∂q x ∂x + ∂q y ∂y + ∂q z ∂z − Q drain = 0 , or, ∇ · q − Q drain = 0 . (7) In view of Eqs. (1) and (4), the discretized weak form of Eq. (7) in an elemental form gives (cid:90) Ω e (cid:16) K B (cid:62) p B p + D N p (cid:62) N p (cid:17) d Ω e (cid:124) (cid:123)(cid:122) (cid:125) A e p e = (cid:90) Ω e D N (cid:62) p p ext d Ω e − (cid:90) Γ e N (cid:62) p q Γ · n e d Γ e (cid:124) (cid:123)(cid:122) (cid:125) f e (8)where, B p = ∇ N p , q Γ represents the Darcy flux through the surface Γ e and N p =[ N , N , N , · · · , N ] are the shape functions for the trilinear hexahedral elements [23]6 y q x dzd y q x dzd y + ( ∂ q x ∂ x dx ) dzd y q y dzdx q y dzdx + ( ∂ q y ∂ y d y ) dzdxq z dxd y q z dxd y + ( ∂ q z ∂ z dz ) dxd y Q drain dzdxx y z Figure 2: A schematic diagram for in- and outflow through an infinitesimal element withvolume dV = dxdydz . Q drain is the volumetric drainage term.used in this paper. For other FEs, Eq. (8) holds similarly with different N p . In a globalsense, Eq. (8) yields Ap = f , (9)where A , p and f are the global flow matrix, pressure vector and loading vector, respec-tively, obtained by assembling their respective elemental terms A e , p e and f e . As p ext = 0and q Γ = 0 are assumed in this work, it follows that f = which leads to Ap = , which issolved with an appropriate input pressure p in boundary condition at a given pressure inletsurface.The obtained pressure field is transformed to a consistent nodal force as [5] F e = − (cid:90) Ω e N (cid:62) u ∇ pd Ω e = − (cid:90) Ω e N (cid:62) u B p d Ω e (cid:124) (cid:123)(cid:122) (cid:125) D e p e , (10)where N u = [ N I , N I , N I , · · · , N I ] with I as the identity matrix in R , and D e repre-senting the elemental transformation matrix. One evaluates the global nodal loads F usingthe following equation F = − Dp , (11)where D is the global transformation matrix which is independent of the design vector. Insummary, the pressure load calculation involves the following 3 main steps:1. Assembly of A , which involves K ( ˜ ρ ) and D ( ˜ ρ ) as design-dependent terms (Eqs. 2, 3, 5, 8)7. Solve Ap = (Eq. 9)3. Calculation of F = − Dp (Eq. 11)Note that step 2 involves a linear system with three times fewer degrees of freedom com-pared to the structural problem, as each node only has a single pressure state. Hence interms of computational cost, the structural analysis remains dominant. This section presents a test problem for illustrating the method and demonstrating theimportance of the drainage term (Eq. 4) in the presented approach. An additional testproblem is included in Appendix 5 to study the effect of flow contrast (cid:15) .Figure 3a depicts the design specifications of the test problem. We consider a domain of L x × L y × L z = 0 . × . × .
01 m , with a pressure load of 1 N m − is applied on the frontface of the domain, and zero pressure on the rear face. The total normal force experiencedby the front face is F x = 10000 × . × .
01 = 10 N. The domain has two solid regions withdimensions L x × L y × L z m , which are separated by L x (Fig. 3). The remaining regionsof the domain are considered void. The design domain is discretized using 48 × × p = N m − p= ×
10 N m
Solid regions a a L (a) (b) (c) Figure 3: (a) Design domain specification to show importance of the drainage term. (b)Pressure field variation (N m − ) without drainage term, (c) Pressure field variation (N m − )with drainage term. Fixed planes are hatched in (a).with and without drainage term (Eq. 9). The pressure and nodal force variations alongthe center of the domain in the x − axis are depicted with and without Q drain in Fig. 4.One notices that if the drainage term is not considered, the pressure gradient does not8 a) Pressure fields (b) Normal forces Figure 4: The pressure field and respective nodal force variations along the x − axis withand without drainage term are depicted. One notices a smooth variation with Q drain ,whereas without the drainage term as expected the pressure field shows a step variationover two solid regions.get confined as soon as the pressure loading faces the first solid region (Figs. 3b and 4a),which is undesirable for the intended purpose. However, a correct behavior is seen whenincluding the drainage term (Figs. 3c and 4b). The corresponding nodal force variationsare also reported in Fig. 4. It is found that the total normal force experienced by the designin Fig. 3c is 10 N which is equal to the original force applied (Fig. 3a). This section presents the optimization problem formulation and the sensitivities of theobjectives with respect to the design vector ρ using the adjoint-variable method. The optimization problem is formulated using a density-based TO framework, whereineach FE is associated with a design variable ρ which is further filtered using the classicaldensity filter [24]. The filtered design variable ˜ ρ i is evaluated as the weighted average of9he design variable ρ j as [24] ˜ ρ i = (cid:80) N e k =1 v j w ( x ) (cid:80) N e k =1 w ( x ) ρ j , (12)where N e represents the total number of FEs lie within the filter radius R fil for the i th element, v j is the volume of the j th element and w ( x ), the weight function, is defined as w ( x ) = max (cid:18) , − || x i − x j || R fill (cid:19) , (13)where || x i − x j || is the Euclidean distance between the i th and j th FEs. x i and x j indicatethe center coordinates of the i th and j th FEs, respectively. The derivative of filtered densitywith respect to the design variable can be evaluated as ∂ ˜ ρ i ∂ρ j = (cid:80) N e k =1 v j w ( x ) (cid:80) N e k =1 w ( x ) . (14)The Young’s modulus of each FE is evaluated using the modified SIMP (Solid IsotropicMaterial with Penalization) formulation as E e ( ρ e ) = E + ˜ ρ eζ ( E − E ) , ˜ ρ e ∈ [0 ,
1] (15)where, E is the Young’s modulus of the actual material, E = 10 − E is set, and thepenalization parameter ζ is set to 3, which guides the TO towards “0-1” solutions.The following topology optimization problem is solved:min ρ f such that: Ap = = F = − DpKv = F d g = V ( ˜ ρ ) V ∗ − ≤ ≤ ρ ≤ , (16)where f is the objective function to be optimized. The global stiffness matrix and dis-placement vector are denoted by K and u , respectively. For designing pressure-loaded 3Dloadbearing structures, compliance, i.e., f = 2 SE is minimized , whereas for the pressure-actuated 3D compliant mechanism designs a multi-criteria [12] objective, i.e., f = − µ MSESE is minimized. SE and M SE represent the strain energy and mutual strain energy of the For loadbearing structure designs, Kv = F d is not considered µ , a scaling factor, is employed primarily to adjust the mag-nitude of the objective to suit the MMA optimizer, and M SE = v (cid:62) Ku is equal to theoutput deformation wherein F d (= Kv ) is the unit dummy force applied in the direction ofthe desired deformation at the output location [12]. Furthermore, V and V ∗ are the actualand permitted volumes of the designs, respectively. We use the gradient-based MMA optimizer [25] for the topology optimization. The adjoint-variable method is employed to determine the sensitivities of the objectives and constraintswith respect to the design variables. One can write an aggregate performance function L for evaluating the sensitivities as L ( u , v , ˜ ρ ) = f ( u , v , ˜ ρ ) + λ (cid:62) ( Ku + Dp ) + λ (cid:62) ( Ap ) + λ (cid:62) ( Kv − F d ) , (17)where λ , λ and λ , the Lagrange multipliers, are determined as [5] λ (cid:62) = − ∂f ( u , v , ˜ ρ ) ∂ u K -1 λ (cid:62) = − λ (cid:62) DA -1 λ (cid:62) = − ∂f ( u , v , ˜ ρ ) ∂ v K -1 . (18)Using the above multipliers (Eq. 18), one can evaluate the objective sensitivities as df d ˜ ρ = ∂f ∂ ˜ ρ + λ (cid:62) ∂ K ∂ ˜ ρ u + λ (cid:62) ∂ A ∂ ˜ ρ p + λ (cid:62) ∂ K ∂ ˜ ρ v , (19)where vectors u , p and v also includes their prescribed values. Now, in view of Eq. 16 andEq. 19, one can subsequently determine the sensitivities for loadbearing structures andCMs with respect to the filtered design vector ˜ ρ as df d ˜ ρ = − u (cid:62) ∂ K ∂ ˜ ρ u + 2 u (cid:62) DA -1 ∂ A ∂ ˜ ρ p (cid:124) (cid:123)(cid:122) (cid:125) Load sensitivities , (20)and df d ˜ ρ = µ MSE ( SE ) (cid:18) − u (cid:62) ∂ K ∂ ˜ ρ u (cid:19) + 1 SE (cid:18) u (cid:62) ∂ K ∂ ˜ ρ v (cid:19) + MSE ( SE ) (cid:18) u (cid:62) DA -1 ∂ A ∂ ˜ ρ p (cid:19) + 1 SE (cid:18) − v (cid:62) DA -1 ∂ A ∂ ˜ ρ p (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) Load sensitivities , (21) A detailed description is given in Ref. [5] omenclature Notation Value Young’s Modulus of a solid FE ( ρ = 1) E × N m − Poisson’s ratio ν . ζ ρ = 0) E E × − N m − External Move limit ∆ ρ p in × N m − K ( ρ ) step location η k K ( ρ ) slope at step β k H ( ρ ) step location η h H ( ρ ) slope at step β h k v N − s − Flow coefficient of a solid FE k s k v × − m N − s − Drainage from solid h s (cid:0) ln r ∆ s (cid:1) K s Remainder of input pressure at ∆ s r 0.1Table 1: Various parameters used in this paper.respectively. Further, one finds the sensitivities of the objectives with respect to the designvector ρ using Eqs. 14, 20 and 21. The load sensitivity terms for both the objectives canbe readily evaluated using Eqs.(20) and (21). As the pressure loads acting on the structuredepend on the design, it is important to include these terms in the optimization. L x x y z L z L y Fixed edgesInput pressure (a) L x x y z L y L z (b) Figure 5: Design domains and problem definitions of the loadbearing structures. (a) Liddomain, (b) Externally pressurized domain.12
Numerical Examples and Discussion
In this section, various small deformation 3D compliant mechanisms actuated via pressureloads and 3D pressure-loaded structures are designed to demonstrate the effectiveness andversatility of the presented approach.Trilinear hexahedral FEs are employed to parameterize the design domains. Optimizationparameters with their nomenclature, symbol and unit are mentioned in Table 1 and anyalteration is reported in the respective problem definition. TO is performed using an in-house MATLAB code with the MMA optimizer. The maximum number of MMA iterationsare set to 100 and 250 for optimizing the loadbearing structures and compliant mechanisms,respectively. The linear systems from state and adjoint equations are solved using theconjugate gradient method in combination with incomplete Cholesky preconditioning.
In this section, two pressure-loaded structure design optimization problems i.e., a lid(Fig. 5a) and an externally pressurized structure (Fig. 5b) are presented. The lid de-sign problem appeared initially in the work of Du and Olhoff [7], whereas the exter-nally pressure structure design problem is taken from Zhang et al. [8]. Let L x , L y and x y z L xL y L z Void region Output location (a) x y z L xL y L z Output locationSP Void region (b) SP SP z L y L z x y z Void region Output location (c)
Figure 6: Design domains and problem definitions of the CMs. (a) Quarter inverter do-main (b) Quarter gripper domain, void non-design region and jaw (green) of the gripper areshown (c) Quarter magnifier domain. Output location for each mechanism design is dis-played, where springs representing stiffness of a workpiece are attached. A void non-designdomain with a rim of solid non-design region around the pressure inlet area is consideredfor each mechanism design, representing the maximum pressure inlet geometry. SP andSP indicate the symmetric planes for the quarter compliant mechanism designs. L z represent the design domain dimensions in x − , y − and z − directions, respectively.13 x × L y × L z = 0 . × . × . is considered for the lid and the externally pressurizeddesign. An inlet pressure p in of 1 bar is applied on the top face of the domains. Edgesdepicted in red are fixed for all design domains (Fig. 5). The permitted volume fraction foreach example is set to 0.25. The lid design is optimized considering the full model , so anytendency of the problem to break the symmetry can be observed. However, the externallypressurized design is optimized by exploiting one of its symmetric conditions, i.e., only halfthe design domain is considered. The full lid and a symmetric half externally pressurizedare parameterized by 120 × ×
60 and 80 × ×
80 hexahedral FEs, respectively. Thefilter radius is set to r min = √ (cid:16) L x N ex , L y N ey , L z N ez (cid:17) for all the solved problems.The optimized designs in different views for the lid and the externally pressurized aredepicted in Fig. 7 and Fig. 8, respectively. To plot the optimized results, an isosurfacewith the physical density value at 0.25 is used. The exterior parts of the optimized designare shown in Fig. 7a and Fig. 8a, respectively. In both cases, the optimizer has succeededin reshaping the pressure-loaded surface into a configuration that is advantageous for theconsidered compliance objective. Material distributions for the optimized lid and externallypressurized loadbearing structures with respect to different cross sections are displayed inFig. 9. One notices that the material densities in the cross sectional planes are close to1.0, which indicates that the optimized designs converge towards 0-1 solutions (Fig. 9).Near boundaries, intermediate densities are seen due to the density filtering. Nevertheless,the results allow for a clear design interpretation. The objectives and volume constraintsconvergence plots are illustrated via Fig. 10a and Fig. 10b, respectively. It is found thatthe convergence plots are smooth and stable. The volume constraint remains active at theend of the optimization for each case and thus, the permitted volume is achieved. (a) (b) (c) (d) (e) Figure 7: The optimized lid design is shown in different view directions. The figure in bindicates the material density color scheme which is kept same for all the solved problems.
Herein, three compliant mechanisms, e.g., inverter, gripper and magnifier are designed in3D involving design-dependent pressure loads using the multi-criteria objective, using the Symmetry conditions are not exploited a) (b) (c) (d) (e) Figure 8: The optimized externally pressurized design is shown in different view directions. (a) (b)
Figure 9: Material distributions of the optimized lid and externally pressurized design withrespect to different cross sections are shown in a and b, respectively. (a) (b)
Figure 10: Objective and volume fraction convergence plots for loadbearing structure prob-lems. (a) Compliance history, and (b) Volume fraction history.formulation given in Eq. 16. These problems have two symmetry planes which are exploitedherein and thus, only quarter of the design domain is optimized for each mechanism.Figure 6a, Fig. 6b and Fig. 6c show the design specifications for one quarter mechanismdesigns. Symmetry planes are also depicted. An inlet pressure load of 1.0 bar is applied15rom the left face of each mechanism design domain, whereas apart from symmetric facesother remaining faces experience zero pressure load. Again as in the previous examples,instead of using a predetermined pressurized surface, the location and shape of the pres-surized structural surface is subject to design optimization using the proposed formulation.Dimensions of each mechanism are set to 0 . × . × . . We use 120 × ×
60 FEs todescribe the considered quarter of each mechanism domain. The permitted volume fractionfor each mechanism is set to 0.1. A rim of solid non-design region with size L x × L y × L z is considered around the pressure inlet area in each quarter mechanism design, indicatingits maximum size. To contain the applied pressure loading, a void non-design domain ofmaximum size L x × L y × L z is considered in front of the loading. The step parame-ters for the flow and drainage coefficients are set to η k = 0 . η h = 0 .
2, respectively[5]. The scaling factor for the objective is set to µ = 100. A unit dummy load is ap-plied along the desired deformation direction of the mechanism to facilitate evaluation ofthe mutual strain-energy. For the quarter gripper design, a jaw (solid passive domain) ofsize L x × L y × L z is considered above a void non-design region with size L x × L y × L z .Each node of the jaw is connected to springs representing the workpiece with a stiffness of50 N m − . The desired gripping motion of the mechanism is in the z − direction. In case ofthe compliant inverter and magnifier mechanisms, the respective workpiece is representedvia springs of stiffness 500 N m − . The desired motion for the inverter mechanism is in thenegative x − direction, whereas for the magnifier an outward movement in the y − directionis sought. (a) (b) (c) (d) (e) Figure 11: Optimized inverter design is shown in different view directions.The symmetric optimized results are transformed into respective final full designs. Fig-ure 11, Fig. 12, and Fig. 13, depict the 3D optimized designs in various views for the com-pliant inverter, gripper and magnifier mechanisms, respectively. The density value of theisosurface is displayed at 0.25. While the TO process produces customized pressure-loadedmembranes, that at the same time act as CMs themselves, the largest part of the designdomain is filled with more or less traditional CM structures, that transmit and convert thepressure-induced deformations into the intended output deformations. In our experiments,we have not found cases where the majority of the design domain became filled with fluid.16 a) (b) (c) (d) (e)
Figure 12: Optimized gripper design is shown in different view directions. (a) (b) (c) (d) (e)
Figure 13: Optimized magnifier design is shown in different view directions. (a) (b) (c)
Figure 14: Material distributions of the optimized inverter, gripper and magnifier compliantmechanisms with respect to different cross sectional planes are displayed in a, b and c,respectively.This is a clear difference from most pressure-loaded active structures as seen in, e.g., thefield of soft robotics, where typically bellows-inspired designs are applied [26]. It is noted,however, that the presented designs are based on linear structural analysis which is only17 a) (b)
Figure 15: Convergence objective and volume fraction plots for compliant mechanisms. (a) − MSESE history, and (b) Volume fraction history. (a) (b) (c)
Figure 16: Deformed profiles are shown with 500 times magnified displacements. Thecolor scheme represents displacement field wherein red and blue indicate maximum andminimum displacements, respectively.valid for a limited deformation range. Note also that the pressurized membranes are notsimply flat but contain corrugations and thicker and thinner regions. Similar to traditionalcompliant mechanisms, these geometries provide preferred deformation patterns that as-sist in the functioning of the mechanisms. The material distributions with respect to thedifferent cross sections for the optimized inverter, gripper and magnifier mechanisms are18llustrated in Fig. 14a, , which show that the structures have converged to clear solid-voiddesigns, within the limits of the applied density filter. The convergence plots for the ob-jectives and volume constraints are illustrated in Fig. 15. One can notice the convergencehistory plots are smooth and stable. The volume constraint for each mechanism design issatisfied and active at the end of the optimization.Figure 16 displays the deformed profiles of the mechanisms. It is seen that in all cases theintended motion is produced. Note that because linear mechanical analysis is used, scalingof deformations is possible within a certain range. To reach deformations comparable to thedesign domain characteristic length, i.e., large deformation, one needs to consider nonlinearmechanics within the topology optimization setting with high pressure loading, which isleft for future research. This also requires configuration-dependent updating of the appliedpressure loads, which could be achieved by solving Eqs. 9 and 11 on the deformed mesh.While the computational cost of these steps is small compared to the deformation analysis,the two problems become bidirectionally coupled and possibly a monolithic approach ispreferred. Also sensitivity analysis of this coupled problem needs further study.
This paper presents a density-based topology optimization approach for designing design-dependent pressure-actuated (loaded) small deformation 3D compliant mechanisms and3D loadbearing structures. The efficacy and versatility of the method in the 3D caseis demonstrated by designing various pressure-loaded 3D structures (lid and externallypressurized design) and pressure-actuated small deformation 3D compliant mechanisms(inverter, gripper and magnifier). For a loadbearing structure, compliance is minimizedwhereas a multi-criteria objective is employed for designing CMs.The Darcy law in association with a drainage term is employed to convert the appliedpressure loads into a design-dependent pressure field wherein the flow coefficient of an FEis related to its design variable using a smooth Heaviside function. It has been illustratedhow the drainage term with the Darcy flux gives an appropriate pressure field for a 3D TOsetting. The presented approach provides a continuous pressure field which is convertedinto consistent nodal forces using a transformation matrix.The method finds pressure loading surfaces implicitly as topology optimization evolves andalso, facilitates easy and computationally cheap evaluation of the load sensitivities usingthe adjoint-variable method. As pressure loading changes its location and magnitude, it isimportant to consider the load sensitivity terms while evaluating the objective sensitivity.The obtained 3D pressure-actuated mechanisms resemble a combination of a tailored pres-surized membrane for load transfer, and a more conventional compliant mechanism designinvolving flexure hinges. It is suggested that different design solutions may emerge once19arger deflections can be included. Extension of the approach with nonlinear continuummechanics is therefore one of the prime directions for future work.
Acknowledgments
The authors would like to gratefully acknowledge Prof. Ole Sigmund for his suggestionsand thanks Prof. Krister Svanberg for providing MATLAB codes of the MMA optimizer.
APPENDIXFlow contrast
Herein, an additional test problems is presented to illustrate the influence of flow contrast (cid:15) using an internally pressurized arc design (Fig. 17a). We consider a 2D setting for the sakeof simplicity and ease of result visualization, but the findings extend naturally to the 3Dcase. The design domain is described via N ex × N ey = 200 ×
100 bi-linear rectangular FEs,where N ex and N ey represent the number of FEs in the x − and y − directions, respectively.The filter radius and volume fraction are set to 2 × min( L x N ex , L y N ey ), and 0.2, respectively. Themaximum number of iterations for the optimization is set to 100. The design parametersmentioned in Table 1 are used.This optimization is performed for a range of (cid:15) values. Fig. 17b depicts the convergencecurve for the compliance objective minimization with the different flow contrasts. Asexamples, Figs. 17c and 17d depict final solutions with respective pressure fields obtainedusing flow contrast (cid:15) = 10 − and (cid:15) = 10 − , respectively. In Fig. 17c it can be seenthat also in void regions, a clear pressure gradient occurs. This is a direct result of thelow flow contrast (Eq. 9). Since a pressure gradient leads to nodal force contributions(Eq. 11), the optimization process creates semi-dense structures to increase the stiffnessof the loaded regions, in order to minimize the total compliance. However, this is not apractical or realistic solution. These artifacts disappear with increased (cid:15) . Based on thisstudy, we recommend that K s K v ∈ [10 − , − ]. In all other numerical examples in thispaper, (cid:15) = 10 − has been used. 20 x = . L y = . p = L y p = p = p = (a) Design domain (b)(c) (cid:15) = 10 − (d) (cid:15) = 10 − Figure 17: (a) Design domain for 2D internally pressurized arc. The optimized results withfinal pressure distribution using (cid:15) = 0 . (cid:15) = 10 − are shown in (c) and (d), respectively. References [1] Sigmund O, Maute K. Topology optimization approaches.
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