Shape optimization is part of the field of
optimal control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and ...
theory. The typical problem is to find the
shape
A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type.
A pl ...
which is optimal in that it minimizes a certain cost
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
while satisfying given
constraints. In many cases, the functional being solved depends on the solution of a given
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
defined on the variable domain.
Topology optimization
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 op ...
is, in addition, concerned with the number of connected components/boundaries belonging to the domain. Such methods are needed since typically shape optimization methods work in a subset of allowable shapes which have fixed topological properties, such as having a fixed number of holes in them. Topological optimization techniques can then help work around the limitations of pure shape optimization.
Definition
Mathematically, shape optimization can be posed as the problem of finding a
bounded set
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
,
minimizing a
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
:
,
possibly subject to a
constraint of the form
:
Usually we are interested in sets
which are
Lipschitz or C
1 boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
and consist of finitely many
components
Circuit Component may refer to:
•Are devices that perform functions when they are connected in a circuit.
In engineering, science, and technology Generic systems
* System components, an entity with discrete structure, such as an assem ...
, which is a way of saying that we would like to find a rather pleasing shape as a solution, not some jumble of rough bits and pieces. Sometimes additional constraints need to be imposed to that end to ensure well-posedness of the problem and uniqueness of the solution.
Shape optimization is an
infinite-dimensional optimization In certain optimization problems the unknown optimal solution might not be a number or a vector, but rather a continuous quantity, for example a function or the shape of a body. Such a problem is an infinite-dimensional optimization problem, becaus ...
problem. Furthermore, the space of allowable shapes over which the optimization is performed does not admit a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
structure, making application of traditional optimization methods more difficult.
Examples
Techniques
Shape optimization problems are usually solved
numerically, by using
iterative method
In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived fr ...
s. That is, one starts with an initial guess for a shape, and then gradually evolves it,
until it morphs into the optimal shape.
Keeping track of the shape
To solve a shape optimization problem, one needs to find a way to represent a shape in the
computer memory
In computing, memory is a device or system that is used to store information for immediate use in a computer or related computer hardware and digital electronic devices. The term ''memory'' is often synonymous with the term ''primary storage ...
, and follow its evolution. Several approaches are usually used.
One approach is to follow the boundary of the shape. For that, one can
sample the shape boundary in a relatively dense and uniform manner, that is, to consider enough points to get a sufficiently accurate outline of the shape. Then, one can evolve the shape by gradually moving the boundary points. This is called the ''Lagrangian approach''.
Another approach is to consider a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
defined on a rectangular box around the shape, which is positive inside of the shape, zero on the boundary of the shape, and negative outside of the shape. One can then evolve this function instead of the shape itself. One can consider a rectangular grid on the box and sample the function at the grid points. As the shape evolves, the grid points do not change; only the function values at the grid points change. This approach, of using a fixed grid, is called the ''Eulerian approach''. The idea of using a function to represent the shape is at the basis of the
level-set method
Level-set methods (LSM) are a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. The advantage of the level-set model is that one can perform numerical computations involving curves and surfaces on a ...
.
A third approach is to think of the shape evolution as of a flow problem. That is, one can imagine that the shape is made of a plastic material gradually deforming such that any point inside or on the boundary of the shape can be always traced back to a point of the original shape in a one-to-one fashion. Mathematically, if
is the initial shape, and
is the shape at time ''t'', one considers the
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
s
:
The idea is again that shapes are difficult entities to be dealt with directly, so manipulate them by means of a function.
Iterative methods using shape gradients
Consider a smooth velocity field
and the family of transformations
of the initial domain
under the velocity field
:
:
,
and denote
:
Then the Gâteaux or shape derivative of
at
with respect to the shape is the limit of
:
if this limit exists. If in addition the derivative is linear with respect to
, there is a unique element of
and
:
where
is called the shape gradient. This gives a natural idea of
gradient descent
In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the ...
, where the boundary
is evolved in the direction of negative shape gradient in order to reduce the value of the cost functional. Higher order derivatives can be similarly defined, leading to Newtonlike methods.
Typically, gradient descent is preferred, even if requires a large number of iterations, because, it can be hard to compute the second-order derivative (that is, the
Hessian
A Hessian is an inhabitant of the German state of Hesse.
Hessian may also refer to:
Named from the toponym
*Hessian (soldier), eighteenth-century German regiments in service with the British Empire
**Hessian (boot), a style of boot
**Hessian f ...
) of the objective functional
.
If the shape optimization problem has constraints, that is, the functional
is present, one has to find ways to convert the
constrained problem into an unconstrained one. Sometimes ideas based on
Lagrange multipliers
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied ex ...
, like the
adjoint state method
The adjoint state method is a numerical method for efficiently computing the gradient of a function or operator in a numerical optimization problem. It has applications in geophysics, seismic imaging, photonics and more recently in neural net ...
, can work.
Geometry parametrization
Shape optimization can be faced using standard optimization methods if a parametrization of the geometry is defined. Such parametrization is very important in CAE field where goal functions are usually complex functions evaluated using numerical models (CFD, FEA,...). A convenient approach, suitable for a wide class of problems, consists in the parametrization of the CAD model coupled with a full automation of all the process required for function evaluation (meshing, solving and result processing).
Mesh morphing
A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands.
Types
* A plastic mesh may be extruded, oriented, ex ...
is a valid choice for complex problems that resolves typical issues associated with
re-meshing such as discontinuities in the computed objective and constraint functions
.
[Wilke, D.N.; Kok, S.; Groenwold, A.A. (2010]
''The application of gradient-only optimization methods for problems discretized using non-constant methods''. Structural and Multidisciplinary Optimization, Vol. 40, 433-451.
/ref>
In this case the parametrization is defined after the meshing stage acting directly on the numerical model used for calculation that is changed using mesh updating methods.
There are several algorithms available for mesh morphing (deforming volume
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies.
* De ...
s, pseudosolids, radial basis function A radial basis function (RBF) is a real-valued function \varphi whose value depends only on the distance between the input and some fixed point, either the origin, so that \varphi(\mathbf) = \hat\varphi(\left\, \mathbf\right\, ), or some other fixed ...
s).
The selection of the parametrization approach depends mainly on the size of the problem: the CAD approach is preferred for small-to-medium sized models whilst the mesh morphing approach is the best (and sometimes the only feasible one) for large and very large models.
The multi-objective Pareto optimization (NSGA II) could be utilized as a powerful approach for shape optimization. In this regard, the Pareto optimization approach displays useful advantages in design method such as the effect of area constraint that other multi-objective optimization cannot declare it. The approach of using a penalty function is an effective technique which could be used in the first stage of optimization. In this method the constrained shape design problem is adapted to an unconstrained problem with utilizing the constraints in the objective function as a penalty factor. Most of the time penalty factor is dependent to the amount of constraint variation rather than constraint number. The GA real-coded technique is applied in the present optimization problem. Therefore, the calculations are based on real value of variables.
See also
* SU2 code
* Topological derivative
*Topology optimization
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 op ...
References
Sources
* Allaire, G. (2002) ''Shape optimization by the homogenization method''. Applied Mathematical Sciences 146, Springer Verlag.
* Ashok D. Belegundu, Tirupathi R. Chandrupatla. (2003) ''Optimization Concepts and applications in Engineering'' Prentice Hall. .
* Bendsøe M. P.; Sigmund O. (2003) ''Topology Optimization: Theory, Methods and Applications''. Springer. .
* Burger, M.; Osher, S.L. (2005) ''A Survey on Level Set Methods for Inverse Problems and Optimal Design''. European Journal of Applied Mathematics, vol.16 pp. 263–301.
* Delfour, M.C.; Zolesio, J.-P. (2001) ''Shapes and Geometries - Analysis, Differential Calculus, and Optimization''. SIAM. .
* Haslinger, J.; Mäkinen, R. (2003) ''Introduction to Shape Optimization: Theory, Approximation and Computation''. Society for Industrial and Applied Mathematic. .
* Laporte, E.; Le Tallec, P. (2003) ''Numerical Methods in Sensitivity Analysis and Shape Optimization''. Birkhäuser. .
* Mohammadi, B.; Pironneau, O. (2001) ''Applied Shape Optimization for Fluids''. Oxford University Press. {{ISBN, 0-19-850743-7.
* Simon J. (1980) ''Differentiation with respect to the domain in boundary value problems''. Numer. Fuct. Anal. and Optimiz., 2(7&8), 649-687 (1980).
External links
Optopo Group
— Simulations and bibliography of the optopo group at Ecole Polytechnique (France). Homogenization method and level set method.
Optimal control
Mathematical optimization