Computer Methods in Applied Mechanics and Engineering
Topology optimization using a mixed formulation: An alternative way to solve pressure load problems
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.
Section snippets
Standard form
We consider an elastic body Ω in equilibrium. The governing equations in strong form for this structure arewhere σij is the symmetric stress tensor, Fi the volume forces, ui the displacement components, ni the surface normal, and Ti are the prescribed displacement and traction forces, respectively, and Γu and ΓT denote the parts of the boundaries of Ω that are controlled by displacement or traction boundary conditions, respectively.
In weak form, the
The topology optimization problem
The goal of a standard topology optimization procedure in linear elasticity [2] is to find the distribution of solid material that minimizes the compliance of a structure. This goal is obtained by introducing a continuous (density) design variable for each element in the structure, assigning an interpolation function that relates the element stiffnesses to the element design variables, performing sensitivity analyses and updating the variables by an optimality criteria algorithm or a math
Examples
This section includes a number of examples intended to demonstrate the idea and efficiency of the method. In the examples, we consider non-dimensionalized properties and dimensions for simplicity, however, the method works just as well for real physical values. In all figures, black denotes solid elastic material and white may denote fluid or void regions depending on the example. In the three-phase cases where confusion is possible, the fluid regions are cross-hatched.
Conclusions
In this paper we have proposed a new method for handling pressure load problems in the density approach to topology optimization. The basic idea of the method is to define the void phase to be incompressible, i.e. as a hydrostatic fluid, and thereby we can transfer the pressure load through the fluid regions. In order to solve problems involving incompressibility we have to define the finite element problem in a mixed displacement/pressure form. This mixed form is a standard method in many
Acknowledgements
The support from the Danish Technical Research Council through the grant “Designing bandgap materials and structures with optimized dynamic properties”, from Eurohorcs/ESF European Young Investigator Award (EURYI) through the grant “Synthesis and topology optimization of optomechanical systems” and from the Danish Center for Scientific Computing (DCSC) is gratefully acknowledged. Also the authors wishes to thank professors Jakob Søndergaard Jensen, Martin P. Bendsøe, Pauli Pedersen and other
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