Topology optimization 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 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 (FEM) to evaluate the design performance. The design is optimized using either gradient-based mathematical programming techniques such as the optimality criteria algorithm and the method of moving asymptotes or non gradient-based algorithms such as genetic algorithms.
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. 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; topology optimization is thus a key part of design for additive manufacturing.
A topology optimization problem can be written in the general form of an optimization problem as:
The problem statement includes the following:
Evaluating often includes solving a differential equation. This is most commonly done using the finite element method since these equations do not have a known analytical solution.
There are various implementation methodologies that have been used to solve topology optimization problems.
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. [1] In literature problems with up to 30000 variables have been reported. [2]
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). [3] [4] 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 . This has been shown to confirm the micro-structure of the materials. [5] In the SIMP method a lower bound on the Young's modulus is added, , to make sure the derivatives of the objective function are non-zero when the density becomes zero. The higher the penalisation factor, the more SIMP penalises the algorithm in the use of non-binary densities. Unfortunately, the penalisation parameter also introduces non-convexities. [6]
There are several commercial topology optimization software on the market. Most of them use topology optimization as a hint how the optimal design should look like, and manual geometry re-construction is required. There are a few solutions which produce optimal designs ready for Additive Manufacturing.
A stiff structure is one that has the least possible displacement when given certain set of boundary conditions. A global measure of the displacements is the strain energy (also called compliance) of the structure under the prescribed boundary conditions. The lower the strain energy the higher the stiffness of the structure. So, the objective function of the problem is to minimize the strain energy.
On a broad level, one can visualize that the more the material, the less the deflection as there will be more material to resist the loads. So, the optimization requires an opposing constraint, the volume constraint. This is in reality a cost factor, as we would not want to spend a lot of money on the material. To obtain the total material utilized, an integration of the selection field over the volume can be done.
Finally the elasticity governing differential equations are plugged in so as to get the final problem statement.
subject to:
But, a straightforward implementation in the finite element framework of such a problem is still infeasible owing to issues such as:
Some techniques such as filtering based on image processing [9] are currently being used to alleviate some of these issues. Although it seemed like this was purely a heuristic approach for a long time, theoretical connections to nonlocal elasticity have been made to support the physical sense of these methods. [10]
Fluid-structure-interaction is a strongly coupled phenomenon and concerns the interaction between a stationary or moving fluid and an elastic structure. Many engineering applications and natural phenomena are subject to fluid-structure-interaction and to take such effects into consideration is therefore critical in the design of many engineering applications. Topology optimisation for fluid structure interaction problems has been studied in e.g. references [11] [12] [13] and. [14] Design solutions solved for different Reynolds numbers are shown below. The design solutions depend on the fluid flow with indicate that the coupling between the fluid and the structure is resolved in the design problems.
Thermoelectricity is a multi-physic problem which concerns the interaction and coupling between electric and thermal energy in semi conducting materials. Thermoelectric energy conversion can be described by two separately identified effects: The Seebeck effect and the Peltier effect. The Seebeck effect concerns the conversion of thermal energy into electric energy and the Peltier effect concerns the conversion of electric energy into thermal energy. [15] By spatially distributing two thermoelectric materials in a two dimensional design space with a topology optimisation methodology, [16] it is possible to exceed performance of the constitutive thermoelectric materials for thermoelectric coolers and thermoelectric generators. [17]
The current proliferation of 3D printer technology has allowed designers and engineers to use topology optimization techniques when designing new products. Topology optimization combined with 3D printing can result in less weight, improved structural performance and shortened design-to-manufacturing cycle. As the designs, while efficient, might not be realisable with more traditional manufacturing techniques.[ citation needed ]
Internal contact can be included in topology optimization by applying the third medium contact method. [18] [19] [20] The third medium contact (TMC) method is an implicit contact formulation that is continuous and differentiable. This makes TMC suitable for use with gradient-based approaches to topology optimization.
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a continuous medium rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.
The Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).
The vorticity equation of fluid dynamics describes the evolution of the vorticity ω of a particle of a fluid as it moves with its flow; that is, the local rotation of the fluid. The governing equation is:
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages. "Finite volume" refers to the small volume surrounding each node point on a mesh.
In continuum mechanics, the Froude number is a dimensionless number defined as the ratio of the flow inertia to the external field. The Froude number is based on the speed–length ratio which he defined as:
In fluid mechanics, or more generally continuum mechanics, incompressible flow refers to a flow in which the material density of each fluid parcel — an infinitesimal volume that moves with the flow velocity — is time-invariant. An equivalent statement that implies incompressible flow is that the divergence of the flow velocity is zero.
Multi-disciplinary design optimization (MDO) is a field of engineering that uses optimization methods to solve design problems incorporating a number of disciplines. It is also known as multidisciplinary system design optimization (MSDO), and multidisciplinary design analysis and optimization (MDAO).
In fluid mechanics, the Taylor–Proudman theorem states that when a solid body is moved slowly within a fluid that is steadily rotated with a high angular velocity , the fluid velocity will be uniform along any line parallel to the axis of rotation. must be large compared to the movement of the solid body in order to make the Coriolis force large compared to the acceleration terms.
Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan and Lucy in 1977, initially for astrophysical problems. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a meshfree Lagrangian method, and the resolution of the method can easily be adjusted with respect to variables such as density.
A crash simulation is a virtual recreation of a destructive crash test of a car or a highway guard rail system using a computer simulation in order to examine the level of safety of the car and its occupants. Crash simulations are used by automakers during computer-aided engineering (CAE) analysis for crashworthiness in the computer-aided design (CAD) process of modelling new cars. During a crash simulation, the kinetic energy, or energy of motion, that a vehicle has before the impact is transformed into deformation energy, mostly by plastic deformation (plasticity) of the car body material, at the end of the impact.
The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, a system of partial differential equations that describe the motion of a fluid in space. Solutions to the Navier–Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering.
Robust optimization is a field of mathematical optimization theory that deals with optimization problems in which a certain measure of robustness is sought against uncertainty that can be represented as deterministic variability in the value of the parameters of the problem itself and/or its solution. It is related to, but often distinguished from, probabilistic optimization methods such as chance-constrained optimization.
The derivation of the Navier–Stokes equations as well as their application and formulation for different families of fluids, is an important exercise in fluid dynamics with applications in mechanical engineering, physics, chemistry, heat transfer, and electrical engineering. A proof explaining the properties and bounds of the equations, such as Navier–Stokes existence and smoothness, is one of the important unsolved problems in mathematics.
In contact mechanics, the term unilateral contact, also called unilateral constraint, denotes a mechanical constraint which prevents penetration between two rigid/flexible bodies. Constraints of this kind are omnipresent in non-smooth multibody dynamics applications, such as granular flows, legged robot, vehicle dynamics, particle damping, imperfect joints, or rocket landings. In these applications, the unilateral constraints result in impacts happening, therefore requiring suitable methods to deal with such constraints.
In fluid dynamics, Airy wave theory gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.
In fluid dynamics, The projection method is an effective means of numerically solving time-dependent incompressible fluid-flow problems. It was originally introduced by Alexandre Chorin in 1967 as an efficient means of solving the incompressible Navier-Stokes equations. The key advantage of the projection method is that the computations of the velocity and the pressure fields are decoupled.
JCMsuite is a finite element analysis software package for the simulation and analysis of electromagnetic waves, elasticity and heat conduction. It also allows a mutual coupling between its optical, heat conduction and continuum mechanics solvers. The software is mainly applied for the analysis and optimization of nanooptical and microoptical systems. Its applications in research and development projects include dimensional metrology systems, photolithographic systems, photonic crystal fibers, VCSELs, Quantum-Dot emitters, light trapping in solar cells, and plasmonic systems. The design tasks can be embedded into the high-level scripting languages MATLAB and Python, enabling a scripting of design setups in order to define parameter dependent problems or to run parameter scans.
The streamline upwind Petrov–Galerkin pressure-stabilizing Petrov–Galerkin formulation for incompressible Navier–Stokes equations can be used for finite element computations of high Reynolds number incompressible flow using equal order of finite element space by introducing additional stabilization terms in the Navier–Stokes Galerkin formulation.
Ole Sigmund is a Danish Professor in Mechanical Engineering who has made fundamental contributions to the field of topology optimization, including microstructure design, nano optics, photonic crystals, Matlab code, acoustics, and fluids. In 2003 he co-authored the highly cited book "Topology Optimization: Theory, Methods and Applications" with Martin P. Bendsøe. His research group was the first to achieve giga-resolution topology optimization, making it for the first time possible to optimize an entire Boeing 777 wing structure.
The third medium contact(TMC) is an implicit formulation for contact mechanics. Contacting bodies are embedded in a highly compliant medium (the third medium), which becomes increasingly stiff under compression. The stiffening of the third medium allows tractions to be transferred between the contacting bodies when the third medium between the bodies is compressed. In itself, the method is inexact; however, in contrast to most other contact methods, the third medium approach is continuous and differentiable, which makes it applicable to applications such as topology optimization.