Arrows indicate increased chamber in the rear wing, which can generate greater downforce. The tool is actually a solver that uses CFD simulation results to find an optimal solution based on stated goals reduced drag, maximized lift-over-draft ratio, reduced pressure drop, etc.

Because it is a solver, it has many advantages:. Because the adjoint solver directly determines which section of the shape to modify and how to do it, it reaches the optimal geometry faster.

- Introduction to Shape Optimization | Society for Industrial and Applied Mathematics.
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Because the adjoint solver works hand-in-hand with mesh morphing technologies, you do not need to redefine the geometry nor recreate the computational mesh; rather you simply morph the mesh to the new shape. In summary, this solution is fast because:. While this is a good approach, it has many limitations: Design shapes can be extremely complex, governed by hundreds of parameters or more. It is impossible to consider all of them. The provided error estimates assess the discretization error with respect to a given quantity of interest and separate the influences of different parts of the discretization time, space, and control discretization.

This allows us to set up an efficient adaptive strategy producing economical locally refined meshes for each time step and an adapted time discretization.

## Topology Optimization | Software And Resources | Autodesk

The space and time discretization errors are equilibrated, leading to an efficient method. This article summarizes several recent results on goal-oriented error estimation and mesh adaptation for the solution of elliptic PDE-constrained optimization problems with additional inequality constraints. The first part is devoted to the control constrained case. Then some emphasis is given to pointwise inequality constraints on the state variable and on its gradient.

In the last part of the article regularization techniques for state constraints are considered and the question is addressed, how the regularization parameter can adaptively be linked to the discretization error. First, general assumptions are stated that allow to prove second-order convergence in control, state and adjoint state.

Afterwards these assumptions are verified for problems where the solution of the state equation has singularities due to corners or edges in the domain or nonsmooth coefficients in the equation. In order to avoid a reduced convergence order, graded finite element meshes are used. Solutions to optimization problems with pde constraints inherit special properties; the associated state solves the pde which in the optimization problem takes the role of a equality constraint, and this state together with the associated control solves an optimization problem, i.

In this note we review the state of the art in designing discrete concepts for optimization problems with pde constraints with emphasis on structure conservation of solutions on the discrete level, and on error analysis for the discrete variables involved. As model problem for the state we consider an elliptic pde which is well understood from the analytical point of view.

This allows to focus on structural aspects in discretization. We discuss the approaches. We consider general constraints on the control, and also consider pointwise bounds on the state. We outline the basic ideas for providing optimal error analysis and accomplish our analytical findings with numerical examples which confirm our analytical results.

## TOPOLOGY OPTIMIZATION

Furthermore we present a brief review on recent literature which appeared in the field of discrete techniques for optimization problems with pde constraints. In this note we present a framework for the a posteriori error analysis of control constrained optimal control problems with linear PDE constraints. It is solely based on reliable and efficient error estimators for the corresponding.

We show that the sum of these estimators gives a reliable and efficient estimator for the optimal control problem. In this article we summarize recent results on a priori error estimates for space-time finite element discretizations of linear-quadratic parabolic optimal control problems. We consider the following three cases: problems without inequality constraints, problems with pointwise control constraints, and problems with state constraints pointwise in time.

For all cases, error estimates with respect to the temporal and to the spatial discretization parameters are derived. The results are illustrated by numerical examples.

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In the first part, we consider Lavrentiev-type regularization of both distributed and boundary control. In the second part, we present a priori error estimates for elliptic control problems with finite-dimensional control space and state-constraints both in finitely many points and in all points of a subdomain with nonempty interior. The treatment of hepatic lesions with radio-frequency RF ablation has become a promising minimally invasive alternative to surgical resection during the last decade. In order to achieve treatment qualities similar to surgical R0 resections, patient specific mathematical modeling and simulation of the biophysical processes during RF ablation are valuable tools.

They allow for an a priori estimation of the success of the therapy as well as an optimization of the therapy parameters.

In this report we discuss our recent efforts in this area: a model of partial differential equations PDEs for the patient specific numerical simulation of RF ablation, the optimization of the probe placement under the constraining PDE system and the identification of material parameters from temperature measurements. A particular focus lies on the uncertainties in the patient specific tissue properties.

We discuss a stochastic PDE model, allowing for a sensitivity analysis of the optimal probe location under variations in the material properties. Moreover, we optimize the probe location under uncertainty, by considering an objective function, which is based on the expectation of the stochastic distribution of the temperature distribution. We present topology optimization of piezoelectric loudspeakers using the SIMP method and topology gradient based methods along with analytical and numerical results. Two injuring effects of cryopreservation of living cells are under study.

First, stresses arising due to non-simultaneous freezing of water inside and outside of cells are modeled and controlled. Second, dehydration of cells caused by earlier ice building in the extracellular liquid compared to the intracellular one is simulated. A low-dimensional mathematical model of competitive ice formation inside and outside of living cells during freezing is derived by applying an appropriate averaging technique to partial differential equations describing the dynamics of water-to-ice phase change.

This reduces spatially distributed relations to a few ordinary differential equations with control parameters and uncertainties.

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Such equations together with an objective functional that expresses the difference between the amount of ice inside and outside of a cell are considered as a differential game. The aim of the control is to minimize the objective functional, and the aim of the disturbance is opposite. A stable finite-difference scheme for computing the value function is applied to the problem. On the base of the computed value function, optimal cooling protocols ensuring simultaneous freezing of water inside and outside of living cells are designed.

Thus, balancing the inner and outer pressures prevents cells from injuring. Another mathematical model describes shrinkage and swelling of cells caused by their osmotic dehydration and rehydration during freezing and thawing. The cell shape is searched as a level set of a function which satisfies a Hamilton-Jacobi equation resulting from a Stefan-type condition for the normal velocity of the cell boundary.

## Shape optimization

Hamilton-Jacobi equations are numerically solved using finite-difference schemes for finding viscosity solutions as well as by computing reachable sets of an associated conflict control problem. Examples of the shape evolution computed in two and three dimensions are presented. The production of nanoscaled particulate products with exactly pre-defined characteristics is of enormous economic relevance.

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- Shape optimal design and free mesh generation | SpringerLink!
- Smart Shape Optimization with the ANSYS Adjoint Solver?

Although there are different particle formation routes they may all be described by one class of equations. Therefore, simulating such processes comprises the solution of nonlinear, hyperbolic integro-partial differential equations. In our project we aim to study this class of equations in order to develop efficient tools for the identification of optimal process conditions to achieve desired product properties.

This objective is approached by a joint effort of the mathematics and the engineering faculty. Two model-processes are chosen for this study, namely a precipitation process and an innovative aerosol process allowing for a precise control of residence time and temperature. Since the overall problem is far too complex to be solved directly a hierarchical sequence of simplified problems has been derived which are solved consecutively.

In particular, the simulation results are finally subject to comparison with experiments. Related Information. Close Figure Viewer. Browse All Figures Return to Figure. Previous Figure Next Figure. Email or Customer ID. Forgot password? Old Password. New Password.

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