Topology optimization of binary structures using integer linear programming

This work proposes an improved method for gradient-based topology optimization in a discrete setting of design variables. The method combines the features of BESO developed by Huang and Xie [1] and the discrete topology optimization method of Svanberg and Werme [2] to improve the effectiveness of binary variable optimization. Herein the objective and constraint functions are sequentially linearized using Taylor's first order approximation, similarly as carried out in [2]. Integer Linear Programming (ILP) is used to compute globally optimal solutions for these linear optimization problems, allowing the method to accommodate any type of constraints explicitly, without the need for any Lagrange multipliers or thresholds for sensitivities (like the modern BESO [1]), or heuristics (like the early ESO/BESO methods [3]). In the linearized problems, the constraint targets are relaxed so as to allow only small changes in topology during an update and to ensure the existence of feasible solutions for the ILP. This process of relaxing the constraints and updating the design variables by using ILP is repeated until convergence. The proposed method does not require any gradual refinement of mesh, unlike in [2] and the sensitivities every iteration are smoothened by using the mesh-independent BESO filter. Few examples of compliance minimization are shown to demonstrate that mathematical programming yields similar results as that of BESO for volume-constrained problems. Some examples of volume minimization subject to a compliance constraint are presented to demonstrate the effectiveness of the method in dealing with a non-volume constraint. Volume minimization with a compliance constraint in the case of design-dependent fluid pressure loading is also presented using the proposed method. An example is presented to show the effectiveness of the method in dealing with displacement constraints. The results signify that the method can be used for topology optimization problems involving non-volume constraints without the use of heuristics, Lagrange multipliers and hierarchical mesh refinement

Topology optimization of binary microstructures involving various non-volume constraints

In this paper, we use the Topology Optimization of Binary Structures (TOBS) method recently developed by Sivapuram and Picelli (2018) for microstructural optimization. This is the first work in topology optimization addressing various non-volume microstructural constraints with discrete (0/1) design variables. The objective and constraint functions are linearized at each iteration, and the obtained linear problem is solved through Integer Linear Programming (ILP) using sensitivities computed from asymptotic homogenization. A periodic filter is used to make the optimized solutions checkerboard-free and mesh-independent. Volume minimization problems subject to elastic and thermal constraints are considered. The examples consider different sets of constraints, including bulk and shear moduli, square/cubic symmetry, isotropy, thermal conductivity and a combination of them in two and three dimensions. The non-volume constraints are treated explicitly, i.e., without the use of Lagrange multiplier/penalty as used in conventional gradient-based binary topology optimization methods (Huang and Xie, 2010). The resulting microstructures are observed to be convergent in all the examples presented and in agreement with the HashinShtrikman bounds

Invited commentary: A patient of pulmonary embolism and online conversational learning among global medical students through a journal review platform

CommentaryVivek Podder, Madhava Sai Sivapuram, Avinash Kumar Gupta, Rakesh Biswa