Topology optimization (TO) is a mathematical method that optimizes material layout within a given design space, for a given set of
loads,
boundary conditions and
constraints with the goal of maximizing the performance of the system. Topology optimization is different from
shape optimization
Shape optimization is part of the field of optimal control theory. The typical problem is to find the shape which is optimal in that it minimizes a certain cost functional while satisfying given constraints. In many cases, the functional being ...
and sizing optimization in the sense that the design can attain any shape within the design space, instead of dealing with predefined configurations.
The conventional topology optimization formulation uses a
finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
(FEM) to evaluate the design performance. The design is optimized using either gradient-based
mathematical programming
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
techniques such as the optimality criteria algorithm and the
method of moving asymptotes or non gradient-based algorithms such as
genetic algorithms
In computer science and operations research, a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to gene ...
.
Topology optimization has a wide range of applications in aerospace, mechanical, bio-chemical and civil engineering. Currently, engineers mostly use topology optimization at the concept level of a
design process
A design is a plan or specification for the construction of an object or system or for the implementation of an activity or process or the result of that plan or specification in the form of a prototype, product, or process. The verb ''to design'' ...
. Due to the free forms that naturally occur, the result is often difficult to manufacture. For that reason the result emerging from topology optimization is often fine-tuned for manufacturability. Adding constraints to the formulation in order to
increase the manufacturability is an active field of research. In some cases results from topology optimization can be directly manufactured using
additive manufacturing
3D printing or additive manufacturing is the construction of a three-dimensional object from a CAD model or a digital 3D model. It can be done in a variety of processes in which material is deposited, joined or solidified under computer co ...
; topology optimization is thus a key part of
design for additive manufacturing
Design for additive manufacturing (DfAM or DFAM) is design for manufacturability as applied to additive manufacturing (AM). It is a general type of design methods or tools whereby functional performance and/or other key product life-cycle considera ...
.
Problem statement
A topology optimization problem can be written in the general form of an
optimization problem
In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions.
Optimization problems can be divided into two categories, depending on whether the variables ...
as:
:
The problem statement includes the following:
* An
objective function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost ...
. This function represents the quantity that is being minimized for best performance. The most common objective function is compliance, where minimizing compliance leads to maximizing the stiffness of a structure.
* The material distribution as a problem variable. This is described by the density of the material at each location
. Material is either present, indicated by a 1, or absent, indicated by a 0.
is a state field that satisfies a linear or nonlinear state equation depending on
.
* The design space
. This indicates the allowable volume within which the design can exist. Assembly and packaging requirements, human and tool accessibility are some of the factors that need to be considered in identifying this space . With the definition of the design space, regions or components in the model that cannot be modified during the course of the optimization are considered as non-design regions.
*
constraints
a characteristic that the solution must satisfy. Examples are the maximum amount of material to be distributed (volume constraint) or maximum stress values.
Evaluating
often includes solving a differential equation. This is most commonly done using the
finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
since these equations do not have a known analytical solution.
Implementation methodologies
There are various implementation methodologies that have been used to solve topology optimization problems.
Discrete
Solving topology optimization problems in a discrete sense is done by discretizing the design domain into finite elements. The material densities inside these elements are then treated as the problem variables. In this case material density of one indicates the presence of material, while zero indicates an absence of material. Owing to the attainable topological complexity of the design being dependent on the number of elements, a large number is preferred. Large numbers of finite elements increases the attainable topological complexity, but come at a cost. Firstly, solving the FEM system becomes more expensive. Secondly, algorithms that can handle a large number (several thousands of elements is not uncommon) of discrete variables with multiple constraints are unavailable. Moreover, they are impractically sensitive to parameter variations. In literature problems with up to 30000 variables have been reported.
Solving the problem with continuous variables
The earlier stated complexities with solving topology optimization problems using
binary variables has caused the community to search for other options. One is the modelling of the densities with continuous variables. The material densities can now also attain values between zero and one. Gradient based algorithms that handle large amounts of continuous variables and multiple constraints are available. But the material properties have to be modelled in a continuous setting. This is done through interpolation. One of the most implemented interpolation methodologies is the Solid Isotropic Material with Penalisation method (SIMP).
a monograph of the subject. This interpolation is essentially a power law
. It interpolates the Young's modulus of the material to the scalar selection field. The value of the penalisation parameter
is generally taken between