Topology optimization using a mixed formulation: An alternative way to solve pressure load problems

Abstract

A shortcoming of the traditional density based approach to topology optimization is the handling of design dependent loads that relate to boundary data, such as for example pressure loads. Previous works have introduced spline and iso-density curves or alternative parametrization schemes to determine the load surfaces. In this work we suggest a new way to solve pressure load problems in topology optimization. Using a mixed displacement–pressure formulation for the underlying finite element problem, we define the void phase to be an incompressible hydrostatic fluid. In this way we can transfer pressure loads through the fluid without any needs for special load surface parametrizations. The method is easily implemented in the standard density approach and is demonstrated to work efficiently for both 2D and 3D problems. By extending the method to a three phase (solid/fluid/void) design method we also demonstrate design of water containing dams.

Introduction

Since its introduction almost two decades ago [1], the topology optimization method for continuum structures [2] has increased tremendously in popularity and is now being used as an everyday mechanical design tool in larger industries and academia all over the world. Also a number of commercial topology optimization tools have been developed, either based on special finite element (FE) solvers or as add-ons to standard commercial FE packages.

While in academia, the applications of topology optimization methods has expanded to a multitude of problems in material design, MicroElectroMechanical Systems (MEMS), fluids, wave-propagation and nano-optics (see [2] for an overview), the most common design problems that are solved by commercial codes are still compliance minimization problems and maximization of lowest eigenfrequency problems with constraints on material resource. Although the solution procedures for these kinds of problems have matured to a satisfactory level, there are still a number of open or less-than-satisfactorily resolved issues with the fundamental topology optimization method. One group of problems concerns inclusion of manufacturing constraints such as minimum and maximum length-scale, draw directions and extrusion constraints; another group concerns inclusion of design dependent loads like pressure load problems.

In pressure load problems, the position of the loads depend on the shape and topology. Such problems are encountered in hydrostatics and dynamics of wind, water and snow loaded mechanical and civil structures such as ships, submerged structures, airplanes, pumps, etc.

A number of papers have addressed the pressure load problems in topology optimization. The Aalborg group [3], [4], [5] has suggested a formulation where the unknown load application curve is determined from an iso-density curve. The sensitivities of the distributed loads with respect to design changes are obtained from an efficient finite difference formulation and a scheme that prevents ill-defined load curves is suggested. In [6] it is suggested to implement the pressure loads by a manipulation of thermal prestrains and a special scheme for identification of “fluid” and void elements is applied. In [7] it is suggested to parametrize a load curve by a spline function and updating element densities and spline parameters in an integrated approach. A phase-field method involving a triple-well function that allows for distribution of three material phases (solid/fluid/void) is introduced in [8]. The scheme includes an explicit penalization of intermediate densities as well as perimeter. In all the above papers, it is stated that the pressure load problem is much more difficult to solve than standard (fixed) load problems. Different more or less elaborate schemes have to be applied in order to update the loading surface. Lately, several applications of the level set method to topology optimization have appeared. Since there exist implicitly given curves describing the boundaries for this method, it is fairly straight forward to implement pressure loads in the level set method [9], [10].

In this paper we introduce a new way to solve the pressure load problem based on a mixed displacement–pressure (incompressible) formulation but using the standard density approach to topology optimization. Thus, the scheme can be implemented in existing softwares based on the density approach. In fact, the only necessary changes to the code lies in the interpolation scheme, the boundary conditions and possibly the linear system solver.

The idea of the method is the following. Instead of defining the equilibrium equations in the typical FE displacement formulation, we define it in mixed form by including the pressure as a separate variable. This makes it possible to define the void phase in the topology optimization formulation as a hydrostatic incompressible fluid, thus allowing for transfer of pressure from the external boundary conditions to the structure – independent of its shape or topology. A potentially weak point of this idea is that internal “void” regions in the structures also become incompressible (fluid-filled) – a possibility the optimization algorithm may take advantage of by using the void regions as “incompressible cavities”. This possibility, that may or may not be physically relevant for a particular problem, can be avoided by introducing an extra (compressible) void phase in the design problem and limiting the volume fraction of the fluid phase. The method thus becomes a 3-phase (solid/fluid/void) topology optimization scheme (see e.g. Refs. [11], [12], [13] for previous work on 3-phase topology optimization). This computational scheme immediately applies to the 2 as well as 3 dimensional cases without further modifications. Also, the idea may be applied to design of water loaded structures like dams or water towers by introducing mass density and gravity loads on the fluid and structural phases.

The paper is organized as follows. In Section 2 we formulate the physical model and compare the standard and mixed formulations for the solving of elasto- and hydrostatic problems. In Section 3 we formulate the topology optimization problem for 2 and 3 phase problems. In Section 4 we demonstrate the method by considering various test problems, some of which are known from the literature. In Section 5 we draw the conclusions and discuss strong and weak points of the method.