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Structural designs proposed by engineers aim to ensure that specific defined requirements are satisfied. Structural optimization is also widely used to identify an admissible design with optimal performance. However, it is important to remember that real mechanical problems exhibit uncertainties in practice that might entail challenges when searching for admissible and/or optimal design solutions. One objective of this paper is to discuss the possible formulations of design problems in this type of uncertain context. By considering uncertainties in the constraint and/or objective functions underlying design problems, several design strategies might in fact be implemented. Especially, this paper should contribute to clarifying the use of the concepts of robustness and reliability in design. After an introduction, the featured design strategies are illustrated and discussed via three academic examples which involve multiple random and design variables. Results show that the structural solutions obtained can be quite sensitive to the formulation of the problem. Thus, the latter should be considered as an essential step during the design procedure. [ABSTRACT FROM AUTHOR]

Numerical layout optimization provides a computationally efficient and generally applicable means of identifying the optimal arrangement of bars in a truss. When the plastic layout optimization formulation is used, a wide variety of problem types can be solved using linear programming. However, the solutions obtained are frequently quite complex, particularly when fine numerical discretizations are employed. To address this, the efficacy of two rationalization techniques are explored in this paper: (i) introduction of 'joint lengths', and (ii) application of geometry optimization. In the former case this involves the use of a modified layout optimization formulation, which remains linear, whilst in the latter case a non-linear optimization post-processing step, involving adjusting the locations of nodes in the layout optimized solution, is undertaken. The two rationalization techniques are applied to example problems involving both point and distributed loads, self-weight and multiple load cases. It is demonstrated that the introduction of joint lengths reduces structural complexity at negligible computational cost, though generally leads to increased volumes. Conversely, the use of geometry optimization carries a computational cost but is effective in reducing both structural complexity and the computed volume. [ABSTRACT FROM AUTHOR]

This paper presents a general mathematical structure for design optimization problems, where state problem functionals are used as design objectives.It extends to design optimization the general model of physical theories pioneered by Tonti (, ) and Oden and Reddy (, ). It turns out that the classical structural optimization problem of compliance minimization is a member of the treated general class of problems. Other particular examples, discussed in the paper, are related to Darcy-Stokes flow and pipe flow models. A main novel feature of the paper is the unification of seemingly different design problems, but the general mathematical structure also explains some previously not fully understood phenomena. For instance, the self-penalization property of Stokes flow design optimization receives an explanation in terms of minimization of a concave function over a convex set. [ABSTRACT FROM AUTHOR]

This paper presents a shape sensitivity analysis and what-if study for two-dimensional multi-scale crack propagation problems using bridging scale decomposition. The sensitivity equations are derived in a continuum setting using direct differentiation method based on a continuum variational formulation of the bridging scale. Due to the fact that the crack propagation speed in an atomistic simulation is discrete in design, and cannot be formulated as a continuous function of shape design variables, we propose a hybrid method that combines analytical sensitivity analysis with finite difference approach. The finite difference part of the sensitivity analysis is only intended for calculating the sensitivity of crack growth speed based on the analytically obtained sensitivity coefficients of structural responses. The theoretical development on sensitivity formulation in this paper extends the application of the method to irregular-shaped finite elements and general design velocity fields. Furthermore, we evaluate and compare several performance measures that quantify crack propagation speed based on crack tip locations for sensitivity analysis and ultimately for structural optimization. A two-dimensional beam example is used to verify the accuracy of the proposed sensitivity approach. It is also demonstrated through a what-if study that with an adequate performance measure, the impact of macroscopic shape changes on microscopic crack propagation speed can be accurately predicted. [ABSTRACT FROM AUTHOR]

This paper presents in-plane material filters for the Discrete Material Optimization method used for optimizing laminated composite structures. The filters make it possible for engineers to specify a minimum length scale which governs the minimum size of areas with constant material continuity. Consequently, engineers can target the available production methods, and thereby increase its manufacturability while the optimizer is free to determine which material to apply together with an optimum location, shape, and size of these areas with constant material continuity. By doing so, engineers no longer have to group elements together in so-called patches, so to statically impose a minimum length scale. The proposed method imposes the minimum length scale through a standard density filter known from topology optimization of isotropic materials. This minimum length scale is generally referred to as the filter radius. However, the results show that the density filter alone gives designs with large measures of non-discreteness. In order to obtain near discrete designs an additional threshold projection filter is applied, so to push the physical design variables towards their discrete bounds. However, because the projection filter is a non-linear function of the design variables, the projected variables have to be re-scaled in a final so-called normalization filter. This is done to prevent the optimizer in creating superior, but non-physical pseudo-materials. The method is demonstrated on a series of minimum compliance examples together with a minimum mass example, and the results show that the method is indeed capable of imposing a minimum length scale onto the optimized layup. [ABSTRACT FROM AUTHOR]

Finding the optimum distribution of material phases in a multi-material structure is a frequent and important problem in structural engineering which involves topology optimization. The Bi-directional Evolutionary Structural Optimization (BESO) method is now a well-known topology optimization method. In this paper an improved soft-kill BESO algorithm is introduced which can handle both single and multiple material distribution problems. A new filtering scheme and a gradual procedure inspired by the continuation approach are used in this algorithm. Capabilities of the proposed method are demonstrated using different examples. It is shown that the proposed method can result in considerable improvements compared to the normal BESO algorithm particularly when solving problems involving very soft material or void phase. [ABSTRACT FROM AUTHOR]

This paper proposes a convenient numerical form-finding method for designing the minimal surface, or the equally tensioned surface of membrane structures with specified arbitrary boundaries. Area minimization problems are formulated as a distributed-parameter shape optimization problem. The internal volume or the perimeter is added as a constraint according to the structure type such as a pneumatic or a suspension membrane. It is assumed that the membrane is varied in the out-of-plane and/or the in-plane direction to the surface. The shape sensitivity function for each problem is derived using the material derivative method. The minimal surface is determined without shape parameterization by the free-form optimization method, a gradient method in the Hilbert space, where the shape is varied by the traction force in proportion to the sensitivity function under the Robin boundary condition. The calculated results show the effectiveness and practical utility of the proposed method for optimal form-finding of membrane structures. [ABSTRACT FROM AUTHOR]

Extending a previous work, a near-optimal family of solutions for the two-load problem is shown, here restricted to the half plain domain. This family is compared with the benchmarks obtained by Sokół and Lewiński (), resulting the cost only slightly larger than the benchmarks, being its geometry simpler. Results taking into account horizontal forces which can be originated by the friction between foundation and ground or by active applied forces are also shown. Finally, an exploration of layouts resulting for solutions to this problem with few nodes is included. [ABSTRACT FROM AUTHOR]