Isogeometric shape optimization for design dependent loads

This paper presents a new isogeometric formulation for shape optimization of structures subjected to design dependent loads. This work considers two types of design dependent loads, namely surface loads like pressure where the direction and/or magnitude of force changes with the variation of boundary shape, and body forces that depend on the material layout. These problems have been mostly solved by topology optimization methods which are prone to difficulties in determination of the loading surface for pressure loads and problems associated with non-monotonous behaviour of compliance and low density regions for body forces. This work uses an isogeometric shape optimization approach where the geometry is defined using NURBS and the control point coordinates and control weights of the boundary are chosen as design variables. This approach accommodates the design dependent loads easily, in addition to its other advantages like exact geometry representation, local control, fewer design variables, excellent shape sensitivity, efficient mesh refinement strategies, and smooth results that can be integrated with CAD. Two classes of optimization problems have been discussed, they are minimum compliance problems subject to volume constraint and minimum weight problems subjected to local stress constraints. These problems are solved using convex optimization programs. Hence, expressions for full sensitivities are derived which is new for structural shape optimization problems with design dependent loads. Some representative engineering examples are solved and compared with existing literature to demonstrate the application of the proposed method.

Isogeometric Shape Optimization for Design Dependent Loads

This paper presents an isogeometric shape optimization approach for a special class of problems in structural optimization known as design dependent load problems. Isogeometric method has been widely used for structural analysis and shape optimization for its advantages in modeling smooth surfaces, high accuracy, local control, and interfacing with CAD tools. Isogeometric method may be of special advantage for shape optimization problems with design dependent loads as the loads in such class of problems depend on the geometry of the designed surface. The method outlined in this paper is applicable to design dependent load problems where there is a pressure load acting on the boundary of the structure, the direction of the pressure being normal to the profile of the designed structure. In this work minimum compliance optimization subject to volume constraint is considered and isogeometric formulations based on NURBS are presented for such class of problems. The control points of the boundaries of the NURBS geometries are taken as the design variables in the optimization problems. Analytical sensitivity formulations are derived for design dependent load problems and compared with numerically derived sensitivities. A few representative 2D and 3D examples are solved and compared with existing literature to demonstrate the application of the proposed method.

Level Set Topology Optimization for Design Dependent Hydrostatic Loading Using the Reproducing Kernel Particle Method

Level set topology optimization for the design of structures subjected to design dependent hydrostatic loads is considered in this paper. Problems involving design-dependent loads remain a challenge in the field of topology optimization. In this class of problems, the applied loads depend on the structure itself. The direction, location and magnitude of the loads may change as the shape of the structure changes throughout optimization. The main challenge lies in determining the surface on which the load will act. In this work, the reproducing kernel particle method (RKPM) is used in combination with the level set method to handle the dependence of loading by moving the particles on the structural boundary throughout the optimization process. This allows for the hydrostatic pressure loads to be applied directly on the evolving boundary. One-way fluid-structure coupling is considered here. A hydrostatic pressure field governed by Laplace’s equation is employed to compute the pressure acting on linear elastic structures. The objective in this optimization problem is to minimize compliance of these structures. Numerical results show good agreement with those in the literature.

A Density Gradient Approach to Topology Optimization Under Design-Dependent Boundary Loading

The paper proposes a density gradient based approach to topology optimization under design-dependent boundary loading. In the density-based topology optimization method, we impose the design dependent loads through spatial gradient of the density. We transform design-dependent boundary loads into a volume form through volume integral of density gradient. In many applications where loadings only need to be exerted on partial boundary, we introduce an auxiliary loading density to keep track of the loading boundary. During the optimization, the loading density is updated by tracking the changes of the physical density in the vicinity of the loading boundary at previous iteration. The proposed approach is easy to implement and computationally efficient. In addition, by adding more auxiliary density fields, the proposed approach is applicable to multiple design-dependent loads. To prevent the intersection of different loading boundaries, a Heaviside projection based integral constraint is developed. Both heat conduction problems under convection loading and elastic problems under hydrostatic pressure loading are presented to illustrate the effectiveness and efficiency of the method.