6 The design of structures using topology optimization can improve the structural performance and save material, 7 in turn reducing costs. Using a framework of large-scale, three-dimensional topology optimization implemented 8 by the authors in an open-source multiphysical software, we investigate the inﬂuence of uncertain loading on the 9 optimized design. Direct differentiation is used to reveal the relationship between displacements and applied force, 10 giving an efﬁcient and effective tool to postprocess optimized topologies. The developed methodology for the 11 assessment of the sensitivity with respect to applied forces is explored using two three-dimensional examples: the 12 classic MBB cantilever and a cableway pylon. The advantages and limitations of this method are discussed. 13 Introduction 14 In the development of new structures and mechanical systems, the proper load collective is critical for correct 15 design and dimensioning. Unforeseen usage, misuse and re-purposing afﬂict the developed load collective with 16 uncertainty and risk. 17 As topology optimization designs structures by ﬁnding the optimally allocated material for a speciﬁc load 18 case or load collective, the uncertainty of loads is especially of interest. In this work, we pose the question, how 19 “optimal” are designs with respect to (w.r.t.) the intrinsic uncertainty of load cases? Using the direct differentiation 20 of the optimized compliance w.r.t. the applied loads, a framework is shown to quickly (and without further system 21 evaluation) assess the inﬂuence of uncertain loading on the optimized design. 22 Topology optimization is a numerical method, which in concert with ﬁnite-element analysis, searches for the 23 optimal allocation of material within a design domain (packaging) for a load case or a set of load cases (load col- 24 lective). Topology optimization has seen recent attention as an essential tool for the Virtuous Circle of Lightweight 25 Engineering Design [1] in which lighter systems result in lowered loading as well as reduced material use and lower 26 costs and therefore resulting in even lighter designs. The advent of additive manufacturing has further strength- 27 ened the attention to this design tool. Additive technologies allow for a practical manufacturing method for the 28 resulting optimized topologies, which in past required elaborate and skilled interpretation to derive manufacturable 29 designs [2]. 30 1 Hofer & Wehrle: The inﬂuence of uncertain loading on topology-optimized designs Uncertainty is a well-established challenge in topology optimization. To handle geometrical uncertainty as- 31 sociated with nano-manufacturing, a robust topology optimization method is developed in [3, 4]. In [5], the latter 32 methodology is extended to ﬁnd not only geometrically robust designs, but also to improve convergence of the 33 optimization algorithm when considering stress constraints. A robust method considering uncertain forces in a 34 level-set framework is introduced in [6]. To consider uncertain loading direction, [7] introduce an interval-based 35 robust topology optimization method. 36 Uncertainty in mechanical engineering design can be categorized into three categories: uncertain geometry, 37 uncertain material and uncertain loading. Further, the modeling itself adds uncertainty via abstraction and simpli- 38 ﬁcation. The aim of this work is to assess the sensitivity of an optimum design w.r.t. uncertain loads and thereby 39 giving feedback to the design engineer how susceptible the optimum design is to changes. The optimization will be 40 carried out using Kratos Multiphysics and its Topology Optimization Application development by [8] as an exten- 41 sion to [9, 10]. The developed methodology is based on the simple direct differentiation applied to the converged 42 solutions and delivers meaningful results regarding the robustness w.r.t. to the load case to the design engineer. 43 Topology optimization 44 Topology optimization solves the design problem of the optimal material use by parameterizing the density, which 45 is then scaled between zero and unity, i.e. no material and full material, 46 xi2[0;1]; (1) wherexiis theith design variable. The design domain is discretized using ﬁnite elements and the density of 47 each element is variable and therefore xiis the density of the ith element. In this work, topology optimization 48 is implemented using the modiﬁed solid isotropic material with penalization (SIMP), ﬁrst introduced by [11] and 49 modiﬁed by [12], 50 Ei=Emin+ (E0Emin)xp i; (2) whereE0is the Young’s modulus for the base material. For this modiﬁcation a minimal Young’s modulus Emin 51 is introduced to avoid singularities in global stiffness matrix. Additionally, the penalty term pprovides a more 52 discrete solution of the problem. Within this work, a penalty term of p= 3is used [13]. 53 Optimization formulation 54 In topology optimization, the optimal material distribution is sought for a deﬁned design domain and load case. For achieving efﬁciently stiff structures, the compliance Cis minimized for a speciﬁed volume fraction vfrac, resulting in the optimization formulation, min x2XfC(x)g; (3) subject togj(x)0; j = 1;:::;a; hk(x) = 0; k = 1;:::;n: Compliance is deﬁned here as the strain energy for a deﬁned load. Therefore, the minimization of compliance 55 formulation can be interpreted as a maximization of the stiffness for a given load. The compliance of the given 56 structure with the design variables xand the force Fcan be calculated as follows: 57 C(x) =FTu: (4) Using the dependency of the displacement uand the global stiffness matrix k, 58 F=ku; (5) 2 Hofer & Wehrle: The inﬂuence of uncertain loading on topology-optimized designs we calculate the compliance on an element basis, shown here for the ith element, 59 Ci(xi) =uT ikiui: (6) The volume fraction vfracis the ratio of the material used, to that of a full design domain V , 60 vfrac=P ixiVi V ; (7) and is deﬁned as an inequality constraint with respect to a prescribed volume fraction vmax, 61 g=X ixiVivmaxV : (8) The modiﬁed SIMP approach is used for the element compliance (6) and summing these up over the elements of the design domain based on the elemental stiffness matrix of the base material of the ith elementk0;i, the modiﬁed optimization problem reads as min xi2X(X iCi(xi)) ; (9) whereCi(xi) = (Emin+xp i(E0Emin))uT ik0;iui; subject tog(x) =X ixiVivfracV 0; and 0xi1: Sensitivity analysis 62 The sensitivities, i.e. gradients, of the design variables w.r.t. the objective and constraint functions are computed and 63 provided to the optimization algorithm. With this information, the design variable values are updated for the next 64 iteration. As topology optimization typically carries more design variables than objective and constraint functions, 65 the adjoint method is used in order to efﬁciently calculate the sensitivities. 66 The compliance sensitivity (6) is calculated for all elements. Applying the adjoint method and the simpliﬁca- 67 tions as well as the modiﬁed SIMP approach, the objective function gradient of every element can be calculated 68 as 69 @C @xi=pxp1 i(E0Emin)uT ik0;iui: (10) The constraint sensitivity too can be calculated for every element. Due to the fact that every element has a 70 constant volume, the constraint function sensitivity can be written as: 71 @g @xi= 1: (11) Sensitivity ﬁltering 72 In order to avoid common problems with topology optimization, such as checkerboard patterns, different ﬁltering 73 methods can be used. In the case of this work, the sensitivity ﬁltering is implemented in the optimization pro- 74 cess in order to avoid such difﬁculties [14]. Checkerboard patterns refer to areas in the design domain that have 75 full and zero density in neighboring elements, giving rise to a structure that looks like a checker- or chessboard 76 (checkerboard patterns can be seen in ﬁg. 8 (a), where a too low ﬁlter radius was used for the ﬁltering process). 77 Checkerboarding results in an unmanufacturable design as well as artiﬁcially (numerically) high stiffness [13]. The 78 ﬁltering of the sensitivities considering the surrounding elements alleviates this issue. Based on the value of these 79 sensitivities and a weighting factor Hi;j, the gradient of the given element is altered. The weighting factor Hi;j80 evaluates the distance from the ith element to the jth neighboring element. 81 3 Hofer & Wehrle: The inﬂuence of uncertain loading on topology-optimized designs x-coordinatey-coordinatermin increasing weighting factor Hi,j→ Figure 1: Filter weighting factor Hi;jwith the ﬁlter radius rmin. The weighting of the surrounding elements of a given element can be seen in ﬁg. 1, where the distance de- 82 termines the inﬂuence of the neighboring element properties. Depending on the ﬁlter radius, a certain number of 83 neighboring elements are considered. Mathematically, this ﬁltering of the sensitivities is given by 84 @C @xi ﬁlt= P jHi;jxj@C=@xj max (xi; )P jHi;j! : (12) The weighting factor is deﬁned by [12] 85 Hi;j= max (0; rmindist(i;j)); (13) whererminrepresents the ﬁlter radius. With the difference of the radius value and the distance between the given 86 elementiand the element from which the sensitivity is considered (described by the index j), the factor can be 87 calculated. 88 Optimization algorithm 89 The optimization problem is solved in this work using the Optimality Criteria (OC) algorithm, which is commonly 90 used in topology optimization [15]. The OC algorithm searches iteratively for the optimal design using the sensi- 91 tivities of objective and constraint functions. A Lagrange multiplier is introduced to handle the constraint function, 92 here volume fraction. This is done by using a bisectional algorithm. 93 The update of the design variable values is given by [16] 94 xi=8 >>< >>:max (0; xim);forxiB imax (0; xim); min (0; xi+m);forxiB imin (1; xi+m); xiB i; else:(14) For this iterative process, the tuning parameters mandare needed, where mrepresents the move limit of the 95 design variable and set here to 0:2. The tuning parameter is introduced for numerical damping purposes and is 96 set here to 0:5[13]. 97 To evaluate the design variable, Biis deﬁned from the optimality condition as: 98 Bi=1 @f=@xi @g=@xi; (15) 4 Hofer & Wehrle: The inﬂuence of uncertain loading on topology-optimized designs with@f=@xibeing the objective function sensitivity deﬁned as the compliance sensitivity @C=@xiand@g=@xithe 99 constraint sensitivity, which has the value of unity for every element. The former is introduced in sec. of which 100 the ﬁltered sensitivities are used (sec. ). The volume constraint sensitivity is constant throughout the optimization 101 and is simply the identity matrix for uniform meshes, as used here [17]. 102 To determine if convergence has been reached, the relative change in the objective function is observed, 103 C(xk)C xk1 C(x0)stop: (16) For the following observations, the convergence criteria was set to stop= 106. Further, there is a heuristic 104 stopping criteria of maximum iterations, which was set to 200 iterations in this investigation. 105 Compliance sensitivity and robustness with respect to force 106 In this work, the variation of the force and the resulting impact on the compliance is investigated. This is carried 107 out in a post-optimization step to assess the consequence in the applied load. By changing the force vector on the 108 structure and optimizing the design domain, the uncertainty in load cases and the resulting optimized topologies 109 are investigated. 110 Directly differentiating (5) w.r.t. the ith forceFigives 111 @k @Fiu+k@u @Fi=@F @Fi; (17) and solving for displacement sensitivity w.r.t. to the force vector reveals 112 @u @Fi=k1@F @Fi@k @Fiu : (18) The compliance sensitivity w.r.t. force is found by applying the chain rule applied to (4), 113 @C @Fi=@F @Fiu+FT k1@F @Fi@k @Fiu : (19) The sensitivity of the stiffness matrix w.r.t. force is zero in linear-elastic analysis. The sensitivity of the force vector w.r.t. to a single force (or multiple forces) is a vector with ones at the point of interest and zero else, denoted here asei. The compliance sensitivity w.r.t. to force simpliﬁes to @C @Fi=eiu+uei; (20) = 2X iui: Summarizing, the compliance sensitivity w.r.t. to force is twice the resulting displacement of the degree of freedom 114 of the force (or forces) in question. 115 Although design robustness can have a wide range of understandings in engineering design, in this work it is 116 deﬁned in line with the following deﬁnitions: 117 Box & Fung (1994): Robustness means insensitivity to variation [18]. 118 Arvidsson & Gremyr (2008): Robust Design Methodology means systematic efforts to achieve insensitiv- 119 ity to noise factors [19]. 120 Taguchi (1986): Robustness is the state where the technology, product, or process perfor- 121 mance is minimally sensitive to factors causing variability [20]. 122 5 Hofer & Wehrle: The inﬂuence of uncertain loading on topology-optimized designs Mathematically, this is therefore deﬁned in this work as the inverse of the sensitivity w.r.t. the uncertainty or noise 123 factors, i.e. here compliance w.r.t. force, 124 RC;Fi=1 j@C=@Fij: (21) It should be noted that the absolute value of the compliance sensitivity is used as robustness measure of how 125 sensitive a structure is and not how it changes . Speciﬁcally, the higher the sensitivity (gradient) of the compliance 126 w.r.t. the force, the higher the resulting variation and the lower the robustness [21]. For the extreme case of inﬁnite 127 robustness, the sensitivity is zero while for zero robustness, the sensitivity is inﬁnity. 128 Results of numerical investigations to uncertain loading 129 In this study, uncertain loading is explored in topology optimization with three-dimensional domains on two exam- 130 ples. The ﬁrst example is the classic MBB cantilever, e.g. [22], in which the out-of-plane dimension is much smaller 131 than the in-plane dimensions, resulting in a relatively thin-walled cantilever. The second example, developed in [8] 132 ﬁnds the optimized topology of a cableway pylon. 133 Moderately thin-walled MBB cantilever 134 The Messerschmitt–Bölkow–Blohm (MBB) cantilever is a classic benchmark example in topology optimization. It 135 stems from a structural problem of the company Messerschmitt–Bölkow–Blohm (now integrated into Airbus SE) 136 in the design of a ﬂoor structure of the Airbus passenger aircraft [23]. Considering the MBB cantilever, the change 137 of the compliance in respect of the changing force is investigated. For this investigation, a three-dimensional MBB 138 cantilever of the dimensions 60204mm (see ﬁg. 2) is used. The given cantilever beam is considered having 139 a generic isotropic material with a Young’s modulus of 1Pa. The ﬁnite-element model has a regular mesh using 1 140 mm element length, resulting in 4800 hexahedral elements and 6408 nodes. 141 Two base load cases will be investigated below: 1) a downward load of F= [0:0;1:0;0:0]TN, and 2) 142 a diagonal load of F= [0:5;0:5;0:5]TN. These base load cases are used for the numerical investigation of 143 uncertain loading. 144 MBB cantilever under downward loading condition 145 The ﬁrst load case foresees a downward (negative y) load. In the following, this load is varied in x,yandzand the 146 compliance sensitivities are calculated. 147 To investigate the optimized design, the load is varied. In different optimization processes the force in y- 148 direction is increased or decreased, whereas in the x- andz-direction a force is added in the positive or negative 149 direction. To avoid to large step sizes and to evaluate also small variations, the change of the force or the added 150 load was 0:1,0:2and0:3N in both positive and negative directions. The resulting optimized topologies can be 151 seen in ﬁg. 4. 152 On the other hand, when investigating the sensitivities, the step sizes where chosen smaller. Due to the fact that 153 a sensitivity is an information of the given condition, the best way to numerically calculate these values is to choose 154 a very low step size. Additionally, the MMB beam is thin walled in the z-direction, which could lead to a sensitive 155 behavior and therefore a smaller step size should improve the results. 156 Fig. 3 shows the optimized compliance w.r.t. the force in x,yandz, respectively. For both variation of force in 157 xandy, the relationship is nearly linear. For the force in z, the compliance increases in both negative and positive 158 directions as this is a increase in the load. In fact this compliance sensitivity shows a quadratic form w.r.t. force (see 159 ﬁg. 3). Further of interest is the compliance sensitivity in x: the compliance increases with a compression of the 160 structure, but with a load in positive x-direction, the compliance decreases. This means that although an increase 161 in the magnitude of the load, the stiffness of the structure can be increased. 162 6 Hofer & Wehrle: The inﬂuence of uncertain loading on topology-optimized designs FdownFdiagonaly xz 60 mm20 mm4 mm Figure 2: Moderately thin-walled MBB cantilever, showing design domain, boundary condition, loading and ex- emplary optimized topology for downward load case and diagonal load case in red and green, respectively. The analytical sensitivities of the compliance w.r.t. the forces used in the diagrams in ﬁg. 3 can be calculated. 163 For this, the displacement is taken at the node where the force is acting on the structure. Therefore the displacement 164 of the degree of freedom of the applied force. The sensitivity is deﬁned simply by twice the displacement (20). 165 For the calculation of the compliance sensitivity, the displacements are considered where the structure is loaded 166 (here a point load), 167 ui=2 64ui;x ui;y ui;z3 75=2 6411:7877 50:348 0:009891913 75mm: The compliance sensitivities calculated via (20) are given by 168 @C @Fi=2 64@C=@Fx @C=@Fy @C=@Fz3 75=2 6423:5754 100:696 0:019783 75mm: It should be noted that force in y-direction is negative and therefore an increase of the force is negative, which 169 changes the sign of the sensitivity from negative to positive. The numerical values of the sensitivities given by the 170 optimization tool and the different force variation can be seen in ﬁg. 5 and they are listed in the table 1. 171 From the compliance sensitivity, we can then calculate the robustness of compliance w.r.t. a change in the deﬁned load case, RC;Fi=2 641 j@C=@Fxj 1 j@C=@Fyj 1 j@C=@Fzj3 75=2 640:042417 0:00993088 50:55613 751 mm: From this result and comparing it with ﬁg. 3, we can see the weakness of this method. The compliance sensi- 172 tivity w.r.t. the out-of-plane force Fzis orders of magnitude smaller than the other forces. In fact, the analytical 173 sensitivity is numerically zero, showing no change. As can be seen in the right graph of ﬁg. 3, this value is correct, 174 though only locally at 0 N. This can be therefore especially problematic in the case, when there is no load present 175 7 Hofer & Wehrle: The inﬂuence of uncertain loading on topology-optimized designs 0:2 0 0:2455055 change in force Fx[N]optimized compliance C[Nmm] 0:2 0 0:220406080 change in force Fy[N]0:2 0 0:26080100120 change in force Fz[N] Figure 3: Optimized compliance w.r.t. to change in applied force for the MBB cantilever, optimization results, —expected from analytical sensitivity, base load of F= [0:0;1:0;0:0]TN. Table 1: Analytical and numerical (using forward differencing) compliance sensitivities w.r.t. to force for varying step sizes for the MBB cantilever with base load of F= [0:0;1:0;0:0]TN. method step size [N]compliance sensitivity [mm] w.r.t. x-forcey-forcez-force analytical - -23.5754 -100.696 -0.01978 numerical0.25 -21.5662 -88.0697 178.9198 0.1 -23.5443 -87.5041 45.6747 0.05 -22.9254 -97.8153 9.57258 0.01 -24.8677 -82.6836 13.5410 0.001 -23.0390 -83.4930 -221.3520 0.0001 2.3999 -39.7399 -2784.6800 in the direction of the sensitivity. To explore this further, we now look at a second load case for which there are 176 non-zero applied forces for the respective compliance sensitivities. 177 MBB cantilever under diagonal loading condition 178 As the lack of out-of-plane loading shows nonlinear relationship when a force is added in this direction, a diagonal 179 load is investigated. The optimized topology of this base load case can be seen in ﬁg. 6. Again a load variation is 180 applied and these results are seen in ﬁg. 7. Although relationship shows some nonlinearity when varying the loads 181 up to60% of the baseline, the trend is well represented. 182 We calculate the compliance sensitivity w.r.t. to the force again based on the displacement. 183 @C @Fi=2 64@C=@Fx @C=@Fy @C=@Fz3 75=2 643:00255 22:3077 951:3613 75mm: This shows agreeance with the numerical results, cf. ﬁg. 7. Based on the sensitivity, we ﬁnd the robustness, RC;Fi=2 641 j@C=@Fxj 1 j@C=@Fyj 1 j@C=@Fzj3 75=2 640:33305 0:0448277 0:001051133 751 mm: 8 Hofer & Wehrle: The inﬂuence of uncertain loading on topology-optimized designs (a)base load case (b)Fx= 0:1N. (c)Fy= 0:1N. (d)Fz= 0:1N. Figure 4: Optimized topology w.r.t. to change in applied force for the MBB cantilever with base load of F= [0:0;1:0;0:0]TN. 1041031021011003;0002;0001;00001;000 step size Fi[N]compliance sensitivity w.r.t force@C @Fi[Nmm N] @C @Fx @C @Fy @C @Fz Figure 5: Numerical sensitivity of compliance w.r.t. force with varying step size for the MBB cantilever, base load ofF= [0:0;1:0;0:0]TN, thin line shows calculated analytical value. 9 Hofer & Wehrle: The inﬂuence of uncertain loading on topology-optimized designs Figure 6: Optimized structure of the MMB cantilever with the diagnoal load case: F= [0:5;0:5;0:5]TN. 0:2 0 0:2242243244 change in force Fx[N]optimized compliance C[Nmm] 0:2 0 0:2235240245250255 change in force Fy[N]0:2 0 0:20200400600 change in force Fz[N] Figure 7: Optimized compliance w.r.t. to change in applied force for the MBB cantilever, optimization results, —expected from analytical sensitivity, base load of F= [0:5;0:5;0:5]TN. We now see that the out-of-plane force is better represented by the analytical value, despite the relatively thin- 184 walled nature of the design space in this direction. As expected, the compliance in out-of-plane load is orders of 185 magnitude more sensitive w.r.t. in-plane force. 186 MBB cantilever with variation of the sensitivity ﬁlter radius 187 The ﬁlter is used on the sensitivities in order to get a better structure and to avoid i.e. checkerboard patterns (cf. sec. ). 188 The size of the radius deﬁnes the area from which surrounding elements are considered (see ﬁg. 1). 189 The impact of the ﬁlter radius is investigated on the optimized compliance and its sensitivities with both the 190 downward and diagonal loading conditions. This was done by optimizing the MBB beam with the different ﬁlter 191 radii and comparing the results in terms of geometry, compliance value and compliance sensitivity. 192 In ﬁg. 8, the impact of the ﬁlter radius on the optimized topology is seen for the downward loading condition. 193 With a ﬁlter size of 1 mm, equivalent to the element size, a signiﬁcant checkerboarding effect is seen. As expected, 194 with increasing ﬁlter radius, the structure becomes more discrete. Fig. 9(a) shows that this results in a signiﬁcant 195 increase in compliance. It is further shown in ﬁg. 9(b) that the topologies have growing sensitivity to force change 196 with larger ﬁlter radii and the corresponding increase in compliance. This leads to the assumption that a more 197 manufacturable and discrete topology is paid for with a less stiff structure. Additionally, the structure possesses a 198 lower robustness w.r.t. changes in the load. 199 The results for diagonal loading ﬁg. 10 for the diagonal load show similar results. The sensitivity with respect 200 to the out-of-plane load Fzis much higher, as expected. 201 10 Hofer & Wehrle: The inﬂuence of uncertain loading on topology-optimized designs (a)rmin= 1 mm (b)rmin= 1:5mm (baseline) (c)rmin= 3 mm (d)rmin= 4 mm (e)rmin= 5 mm Figure 8: Optimized topology w.r.t. to ﬁlter radius with base load of F= [0:0;1:0;0:0]TN. The optimized structure was postprocessed by which the elements were deleted with density xi<0:2. The increasing compliance with the increasing ﬁlter radius can be explained as follows: We are paying for a 202 more continuous structure (i.e. lack of checkerboarding) with a higher compliance. With an ever larger ﬁlter radius, 203 this phenomenon is further increased. The resulting topology is less branched and more compact. But the price is 204 that the structure loses stiffness. 205 The above effect can be seen analogously for the sensitivities. As seen in the sec. , the compliance sensitivity 206 of the structure is computed with the displacement in the degree of freedom of the load in question (or the sum 207 therefore). With less structural stiffness, the point will be further deformed, leading to an increase in its sensitivity. 208 Cableway pylon 209 The investigation of the force variation was extended to the large-scale example of a cableway pylon. Topology 210 optimization of such tower-like structures can be performed via ground structure [24] and the above described 211 SIMP approach. The former method has been successfully applied to tower structures, e.g. [25–27], though dictates 212 a design of lattice like members. The SIMP approach has seen use for tower-like structures [26, 28] and will be 213 used here. 214 An example of such a cableway pylon is shown in ﬁg. 11(a). The according parts of the cableway pylon can be 215 seen in ﬁg. 11(b). The model was constructed such that the area on the ground is ﬁxed and therefore no displace- 216 ments in this area are allowed. The design domain allows for the gondola to pass the pylon without any problems. 217 The forces onto the pylon are spread over the area on top of the pylon (see red area in ﬁg. 11(c)), where the design 218 domain is connected with the yoke (see ﬁg. 11(b)). For the investigation of the cableway pylon, steel was used and 219 the Young’s modulus was set to 206900 MPa. 220 Due to different load cases and varying situations (gondola ascent or descent, wind in the driving direction 221 or perpendicular to the driving direction etc.), which can occur within the lifetime of the cableway pylon, the 222 resulting force working onto the structure has components in every direction. A simpliﬁed load case was developed, 223 11 Hofer & Wehrle: The inﬂuence of uncertain loading on topology-optimized designs 2 450607080 ﬁlter radius rmin[mm]optimized compliance C[Nmm] (a)Compliance w.r.t. ﬁlter radius.2 4050100150 ﬁlter radius rmin[mm]compliance sensitivity w.r.t force@C @Fi[Nmm N] @C @Fx @C @Fy @C @Fz (b)Compliance sensitivity w.r.t. ﬁlter radius. Figure 9: Inﬂuence of ﬁlter radius for the MBB cantilever with a base load of F= [0:0;1:0;0:0]TN. evaluating the different load cases and choosing a worst case (in every direction). These loads were set for the 224 optimization process in order to cover all possible load combinations: 225 Fapplied =2 6460 115 4003 75kN: The symmetry of the loads was recognized and a quarter model is taken for the optimization. The results 226 are shown as a full model by mirroring the results over the axes of symmetry. The resulting quarter model is 227 composed of 97152 hexahedral elements and 106225 nodes. The model consists of hexahedral elements of the size 228 0:50:50:5m. 229 The force variation for the investigation is realized by varying the existing forces in the main directions. For 230 this the forces are increased or decreased by 1%or0:5%. 231 The variation resulted in different compliance values of the structure. In ﬁg. 12 the resulting changes of the 232 forces can be seen. The numerical results show excellent agreement with the analytical sensitivities. This is the 233 case as we are not assessing the sensitivities at unloaded degrees of freedom as with the MBB cantilever. Further, 234 no dimension can be considered relatively thin-walled, in contrast to the MBB case. 235 The analytical sensitivities of the optimized cableway pylon can be calculated with the displacement at the 236 nodes where the force is working on the structure. In the case of the pylon, the force is working on 25 nodes spread 237 over the area on the top. For the sensitivity the displacements in the force direction have to be added up and then 238 multiplied by two. This results in the following sensitivities: 239 @C @Fi=2 64@C=@Fx @C=@Fy @C=@Fz3 75=2 640:0197291 0:0838628 0:00744533 75m: In ﬁg. 12 the compliance change calculated with the sensitivities is compared with the analytical results. It can be 240 seen that with an increase in the force, the compliance increases too. This means higher forces lead to a structure 241 with lower stiffness. 242 12 Hofer & Wehrle: The inﬂuence of uncertain loading on topology-optimized designs 2 4250300 ﬁlter radius rmin[mm]optimized compliance C[Nmm] (a)Compliance w.r.t. ﬁlter radius.2 405001;000 ﬁlter radius rmin[mm]compliance sensitivity w.r.t force@C @Fi[Nmm N] @C @Fx @C @Fy @C @Fz (b)Compliance sensitivity w.r.t. ﬁlter radius. Figure 10: Inﬂuence of ﬁlter radius for the MBB cantilever with a base load of F= [0:5;0:5;0:5]TN. (a) Exemplary pylon structure of Ritten’s cableway (South Tyrol, Italy), Photograph: STA/Riller. (b) Schematic view with the yoke (1) upon which the shoes or sheave assembly (2) are mounted, on which in turn the gondola (3) runs to pass the pylon. The yoke itself is mounted onto the design domain (4). The designable domain is shown in light gray. x yz (c)Quarter model used in topol- ogy optimization with design do- main shown in gray and loaded surface in red. Figure 11: Cableway pylon. 13 Hofer & Wehrle: The inﬂuence of uncertain loading on topology-optimized designs 20 0 20268:5269 change in force Fx[N]optimized compliance C[Nm] 4020 0 20 40266268270272 change in force Fy[N] 100 0 100268269270 change in force Fz[N] Figure 12: Optimized compliance w.r.t. to change in applied force for the cableway pylon, optimization results, —expected from analytical sensitivity, base load of F= [60000;115000;400000]TN. The robustness of the compliance w.r.t. loading is the inverse of the compliance sensitivity resulting in RC;Fi=2 641 j@C=@Fxj 1 j@C=@Fyj 1 j@C=@Fzj3 75=2 6450:6865 11:9242 134:31293 751 m: Through this assessment we can easily see that the resulting design for the cableway pylon is most robust w.r.t. to 243 a downward load, i.e. an axial load, while it is more sensitive in bending, as one may expect. Therefore, especially 244 these loads must be carefully derived in the planning of such a structure. 245 Conclusion 246 The deﬁnition of loads, i.e. the loads that a structure will see in its life, is afﬂicted by uncertainty. As topology 247 optimization designs a structure based on its loading, the proper deﬁnition of load collectives is therefore essential 248 to gain useful results. In this work, we derive a methodology to calculate the sensitivity of the resulting compliance 249 w.r.t. to the change in the load case without any further evaluations. This simple, yet useful method gives the 250 designing engineer feedback to the optimized design and the sensitivity to changes in loading. 251 This method though shows limitation when the structure is relatively thin walled, i.e. one dimension is of a 252 smaller order of magnitude and unloaded in this direction. As is shown with the MBB cantilever, when unloaded, 253 14 Hofer & Wehrle: The inﬂuence of uncertain loading on topology-optimized designs the out-of-plane force shows zero sensitivity. In the numerical studies, a new out-of-plane load greatly changes the 254 resulting optimized topology and the sensitivity is by no means zeros. With the cableway pylon, this aspect is not 255 a problem, showing that the thin-walled case is a special case and in need of further investigation. 256 The investigation of the ﬁlter radius shows that with an increasing ﬁlter radius a more manufacturable topology 257 is achieved. The more concrete the structure gets the lower the stiffness will be. Higher compliance sensitivities 258 w.r.t. the load are to be expected with a higher ﬁlter radius. This means that even though the structure has an easier 259 topology we lose structural properties. Therefore a compromise has to be made when choosing the ﬁlter radius. 260 This method is not only applicable to compliance-minimizing topology optimization resulting in stiff structures, 261 but to the design of compliant mechanisms, which are designed to have large deformations in certain degrees of 262 freedom (e.g. [5, 29, 30]. With such designs, the design of the force and therefore the compliance sensitivity w.r.t. 263 force is of great concern. 264 Summarizing, this work provides a method requiring no further computational effort to assess the compliance 265 sensitivity of topology optimized designs. Although, weaknesses exist in regards to compliance sensitivity w.r.t. 266 loads that were not considered, its is meaningful to assess the robustness, to compare sensitivities of different loads 267 and the sensitivity of a structure to loading in general. 268 Data Availability 269 The data used to support the ﬁndings of this study are available from the corresponding author upon request. 270 Conﬂicts of Interest 271 The authors declare that they have no conﬂicts of interest. 272 Acknowledgements 273 This work is supported by the project RTD 2020 – TN201Q LighOpt Lightweight engineering of multibody systems 274 with design optimization funded by the Free University of Bozen-Bolzano. The authors would like to thank Joseph 275 Reinisch at Leitner AG (Sterzing, South Tyrol, Italy) for providing the load case for the cableway pylon problem. 276 A preprint of this article has previously been published in [31]. 277