A qualitative evaluation and structural analysis of multiple and additive load cases for two-dimensional Multi-Material Topology Optimisation in Grasshopper using the Generalised SIMP method

Multi-cell thin-walled structures exhibit significant advantages in maximizing energy absorption and minimizing mass during vehicle crashes. Since the topological distribution of wall members has an appreciable effect on the crashworthiness, their design signifies an important area of research. As a major energy absorber, multi-cell tubes are more commonly encounter oblique loading in real life. Thus, this study aimed to optimize multi-cell cross-sectional configuration of tubal structures for multiple oblique loading cases. An integer coded genetic algorithm (ICGA) is introduced here to optimize topological distribution of multi-celled web members for single/multiple oblique impacting conditions. Specifically, material distribution in a form of allocating web wall thickness, starting from zero, is considered as design variables and maximization of energy absorption (EA) as the design objective under the predefined peak crushing force and structural mass constraints. The optimization allows generating uniform or non-uniform thickness distribution in different web wall configurations to maximize usage efficiency of material. Compared with the baseline structure, the optimized configurations largely improved the energy absorption in both single and multiple load cases. The examples demonstrate that the proposed ICGA-based design method not only provides a useful approach to searching for novel crashworthy structures in a systematic fashion, but also develops a series of novel multi-cell topologies for multiple oblique loading cases.

Despite significant improvements in terms of the predictive ability of Quantitative Computed Tomography based Finite Element (QCT-FE) models in estimating femoral strength (fracture load and stiffness), no substantial clinical adoption of this method has taken place to date. Narrowing the wide variability of FE results by standardizing the methodology and validation protocols, as well as reducing the uncertainties in the FEA process have been proposed as routes towards improved reliability. The aim of this study was to: First, validate a QCT-FE model of proximal femoral stiffness in multiple stance load cases, and second, using a parametric approach, determine the influence of select experimental and modeling parameters on the predictive ability of our model. Ten fresh frozen human femoral samples were tested in neutral stance, 15° adducted and 15° abducted load cases. Voxel-based linear-elastic QCT-FE models of the samples were generated to predict the models’ stiffness values in all load cases. The base FE models were validated against the experimental results using linear regression. Thirty six deviated models were created using the minimum and maximum values of experiment-based “plausible range” for 18 parameters in 4 categories of embedding, loading, material, and segmentation. The predictive ability of the models were compared in terms of the coefficient of determination (R2) of the linear regression between the measured and predicted stiffness values in all load cases. Our model was capable of capturing 90% of the variation in the experimental stiffness of the samples in neutral stance position (R2 = 0.9, concordance correlation coefficient (CCC) = 0.93, percent root mean squared error (RMSE%) = 8.4%, slope and intercept not significantly different from unity and zero, respectively). Embedding and loading categories strongly affected the predictive ability of the models with an average percent difference in R2 of 4.36% ± 2.77 and 2.96% ± 1.69 for the stance-neutral load case, respectively. The performance of the models were significantly different in adducted and abducted load cases with their R2 dropping to 71% and 70%, respectively. Similarly, off-axes load cases were affected by the parameters differently compared to the neutral load case, with the loading parameter category imposing more than 10% difference on their R2, larger than all other categories. We also showed that automatically selecting the best performing plausible value for each parameter and each sample would result in a perfectly linear correlation (R2> 0.99) between the “tuned” model’s predicted stiffness and experimental results. Based on our results, high sensitivity of the model performance to experimental parameters requires extra diligence in modeling the embedding geometry and the loading angles since these sources of uncertainty could dwarf the effects of material modeling and image processing parameters. The results of this study could help in improving the robustness of the QCT-FE models of proximal femur by limiting the uncertainties in the experimental and modeling steps.

This paper utilizes the TIMP (Transplanting ICM Ideas into Material with Penalization) method to solve topology optimization problems of plate structures with displacement constraints under multiple loading cases. Two basic perspectives embedded within the TIMP method are that, (1) one more penalty function is added besides the Young's modulus penalty function by transplanting the ICM ideas and progresses into the SIMP (Solid Isotropic Material with Penalization) method, and (2) the definition of the artificial material design variables, as well as the idea of penalization on materials, is inherited from the SIMP method. Based on the TIMP method, complex topology optimization problems with displacement constraints are expressed explicitly by using the element weight and Young's modulus penalty functions, and the nonlinear programming algorithm is used to get solutions. Displacement filtering is employed to eliminate the mesh-dependency and checkerboard issues, and the coarse selection of quasi-active constraint strategy is adopted to select active constraints and improve the computing efficiency. The whole solution development process is implemented into a secondary development software based on the Abaqus software by its script language Python. Two problems with a single loading case and two problems with multiple loading cases are addressed on this secondary development software. The effects of using linear and nonlinear element weight penalty functions on the convergence speed are observed through these numerical problems. The results demonstrate that the TIMP method is effective to undertake complex topology optimization problems with displacement constraints under multiple loading cases.

On design of novel bionic bamboo tubes for multiple compression load cases

A modified interior penalty algorithm for the optimization of structures subjected to multiple independent load cases

Modified Rejection Ratio for Multiple Load Cases Evolutionary Structural Optimization

This paper concerns the heuristic-based material stiffness optimization of frictional linear elastic contact problems for having control over the contact stress distribution, aiming to extend the material stiffness optimization to multiple loading conditions, in which each of the loadings acts solely on the structures. A decrease level of the variance of the contact stress is introduced and a weighted sum of the decrease levels under all load cases is constructed as the objective function. The individual criterion for contact problems with multiple contact regions is addressed. The worst case design is adopted for multiple load cases, and an extreme reference stress, which is the highest stress level of the subdomain under all load cases, is defined to control the Young’s modulus modification process in a finite element framework. Through three numerical examples, it is demonstrated how an even distribution of the contact stress can be obtained for contact problems subjected to multiple load cases with single or multiple contact regions. Some new features of the material stiffness optimization with multiple loading conditions are also illustrated.

By choosing the density of particle as the design variables, a new implementation method of topology optimization is presented based on the Element-free Galerkin (EFG) method in this paper, in which the optimal objective is to minimize structural compliance. The advantage of using nodal density is that the displacement and density in the influence domain have the same approximation scheme, and the smoothness of the density field can be improved. Nodal density method presented can prevent the checkerboard from the mathematical model proposed. The topology optimal model based on EFG method under multiple loading cases and stress constraints is proposed, and the sensitivity analysis of optimal design is derived in detail. By using solid isotropic material with penalization (SIMP) method and optimality criteria (OC) method, an algorithm of topology optimization based on the EFG method is presented. The shortcoming of using nodal density can be overcome by introducing the penalty function method. Three topology optimal examples are solved successfully and test the model and algorithm proposed. The results obtained show that the checkerboard phenomenon arisen in topology optimization is not found, and the method proposed is not only effective in suppressing checkerboards but also has better convergence.

This article proposes a new topology optimization method for the design of structures under multiple loading cases. The design is formulated as a multi-objective optimization problem by minimizing a new compliance–volume product, which optimizes the overall stiffness and volume simultaneously to avoid the empirical decision on design constraints and obtain an even lower structural volume. A normalized exponential weighted criterion (NEWC) method is included in the multi-objective optimization problem for the capture of the entire Pareto frontier. A weight evaluation method, in terms of the fuzzy multiple-attribute group decision-making (FMAGDM) theory, is incorporated in the problem to evaluate the weights of the objectives and guarantee the optimal design in an acceptable level. The solid isotropic material with penalty (SIMP) method is used to represent the dependence of elemental densities on material properties. Three typical numerical examples are employed to show the effectiveness of the proposed method.

This paper presents a method to optimize the topology of structures under multiple load cases with stress constraints. Fiber-reinforced orthotropic composite is employed as the material model to simulate the constitutive relation of truss-like continua. The fiber densities and orientations at the nodes are taken as design variables. First, for each load case, the fiber orientations are aligned with the orientations of principal stress and the fiber densities are adjusted according to the strains along the fiber orientations. Then, to optimize the structure, the fiber densities and orientations under multiple load cases are determined by constraining its elastic matrix to approach the elastic matrix of the optimum structures defined for each single load case. Finally the member distribution in the optimal structure is suggested by the continuous lines formed according to the fiber densities and orientations. Several examples are presented to demonstrate the effectiveness of the proposed approach.

This paper is dedicated to designing the overall structural topology for the lightweight design of an automobile wheel. A simplified two-dimensional finite element analysis (FEA) model for the wheel is established, in which the whole wheel structure is first defined as design domain during topology optimization. A rotationally periodic constraint is introduced to design the wheel into structural topology consisting of rotationally repetitive modules. Further, compliance-based topological design under multiple load cases within single module is carried out. In order to achieve a uniform deflection and stiffness distribution around the circumference of wheel, a weighted compliance under multiple load cases is taken as the objective function. In addition, some factors significantly affecting the structural topology are discussed.

A new framework, EBF3PanelOpt, is being developed for design and optimization of complex, multifunctional, aircraft structural concepts like pressurized non-circular fuselage structures to be used in hybrid wing/body vehicles and subjected to complex structural loading cases. This framework can be used to integrate materials and structural concepts to exploit emerging additive manufacturing processes that offer the ability to efficiently fabricate complex structural configurations. Commercial software packages, Msc.Patran (geometry modeling and mesh generation), Msc.Nastran (Finite Element Analysis), VisualDoc (external optimizer) are integrated in EBF3PanelOpt framework using Python programming environment to design stiffened panels with curvilinear stiffeners. Typically, these panels experience multiple loading conditions during the operations of these vehicles. So EBF3PanelOpt has been enhanced to optimize flat/curved multi-sided panels with straight/curved edges having curvilinear, blade-type stiffeners under multiple loading conditions. The mass of the panel is minimized subjected to constraints on buckling, von Mises stress, and crippling or local failure of the stiffener using both global optimization techniques and gradient based optimization techniques. The panel/stiffener geometry is defined in a parametric fashion based on design variables that include variables for orientation and shape of the stiffeners, the thicknesses and height of the stiffeners, and the plate thickness. During optimization, constraints can be applied for each of the loading conditions by aggregating all the responses using Kreisselmeir-Steinhauser criteria or using worst response amongst all the responses or applying all the constraints. In this paper, the optimization of flat rectangular and cylindrical panel will be carried out for three sample load cases of practical interest. This paper discusses the advantages and disadvantages of the proposed constraints’ application.

This paper proposes a method to optimize the reinforcement layout of RC structures under multiple load cases (MLCs) using the planar truss-like material model. It is assumed that concrete is filled with truss-like materials. Two families of orthotropic members in the truss-like materials are used to simulate steel bars. The densities and orientations of steel bars at nodes are considered as design variables. The optimization problem is to minimize the total volume of steel bars with stress constraints. First, under each load case, the distribution of steel bars is optimized as per the fully stressed criterion. Second, based on the results obtained above, the directional stiffness of steel bars under MLCs, described by a closed quadratic curve, is determined using the least squares method. Finally, by solving the eigenvalues problem of the coefficient matrix of the quadratic curve, the optimal distribution of steel bars under MLCs is obtained.

A meshless method for topology optimization of structures under multiple load cases

In this paper, a topology optimization method based on the element independent nodal density (EIND) is developed for continuum solids with multiple load cases and multiple constraints. The optimization problem is formulated ad minimizing the volume subject to displacement constraints. Nodal densities of the finite element mesh are used a the design variable. The nodal densities are interpolated into any point in the design domain by the Shepard interpolation scheme and the Heaviside function. Without using additional constraints (such ad the filtering technique), mesh-independent, checkerboard-free, distinct optimal topology can be obtained. Adopting the rational approximation for material properties (RAMP), the topology optimization procedure is implemented using a solid isotropic material with penalization (SIMP) method and a dual programming optimization algorithm. The computational efficiency is greatly improved by multithread parallel computing with OpenMP to run parallel programs for the shared-memory model of parallel computation. Finally, several examples are presented to demonstrate the effectiveness of the developed techniques.