http://www.cmap.polytechnique.fr/~jouve/papers/jcp_final.pdf

Structural optimization using sensitivity analysis and a level-set method

Topology optimization of continuum structures: A review*

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

History matching is a type of inverse problem in which observed reservoir behavior is used to estimate reservoir model variables that caused the behavior. Obtaining even a single history-matched reservoir model requires a substantial amount of effort, but the past decade has seen remarkable progress in the ability to generate reservoir simulation models that match large amounts of production data. Progress can be partially attributed to an increase in computational power, but the widespread adoption of geostatistics and Monte Carlo methods has also contributed indirectly. In this review paper, we will summarize key developments in history matching and then review many of the accomplishments of the past decade, including developments in reparameterization of the model variables, methods for computation of the sensitivity coefficients, and methods for quantifying uncertainty. An attempt has been made to compare representative procedures and to identify possible limitations of each.

Recent progress on reservoir history matching: a review

This review paper provides an overview of different level-set methods for structural topology optimization. Level-set methods can be categorized with respect to the level-set-function parameterization, the geometry mapping, the physical/mechanical model, the information and the procedure to update the design and the applied regularization. Different approaches for each of these interlinked components are outlined and compared. Based on this categorization, the convergence behavior of the optimization process is discussed, as well as control over the slope and smoothness of the level-set function, hole nucleation and the relation of level-set methods to other topology optimization methods. The importance of numerical consistency for understanding and studying the behavior of proposed methods is highlighted. This review concludes with recommendations for future research.

Level-set methods for structural topology optimization: a review

The aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole. In this paper, such an expansion is obtained and analyzed in the context of linear elasticity for general functionals and arbitrary shaped holes by using an adaptation of the adjoint method and a domain truncation technique. The method is general and can be easily adapted to other linear PDEs and other types of boundary conditions.

The Topological Asymptotic for PDE Systems: The Elasticity Case

The topological sensitivity analysis consists in study- ing the behavior of a shape functional when modifying the topology of the domain. In general, the perturbation under consideration is the creation of a small hole. In this paper, the topological asymp- totic expansion is obtained for the Laplace equation with respect to the insertion of a short crack inside a plane domain. This result is illustrated by some numerical experiments in the context of crack detection.

/pdf/crack-detection-by-the-topological-gradient-method-junwwzndx8.pdf

Crack detection by the topological gradient method

The aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a functional with respect to the creation of a small hole in the domain. In this paper such an expansion is obtained for the Helmholtz equation with a Dirichlet condition on the boundary of a circular hole. Some applications of this work to waveguide optimization are presented.

The Topological Asymptotic for the Helmholtz Equation

The aim of the topological sensitivity analysis is to derive an asymptotic expansion of a design functional with respect to a topological perturbation of the domain. In this paper, such an expansion is obtained for the 3D Maxwell equations when we introduce a small dielectric object in the domain and when we insert a small metallic obstacle. Some numerical results are presented in the context of buried object detection and shape inversion of 3D objects in free space from time-domain scattered field data.

https://iopscience.iop.org/article/10.1088/0266-5611/21/2/008/pdf

The topological asymptotic expansion for the Maxwell equations and some applications

In shape optimization problems, each computation of
the cost function by the finite element method
leads to an expensive analysis. The use of the second order derivative
can help to reduce the number of analyses. Fujii ([4], [10])
was the first to study this problem. J. Simon [19] gave the second order
derivative for the Navier-Stokes
problem, and the authors describe in [8], [11], a method which gives an
intrinsic expression of the first and second order derivatives on the
boundary
of the involved domain.

In this paper we study higher order derivatives. But one can ask
the following questions:

-- are they expensive to calculate?

-- are they complicated to use?

-- are they imprecise?

-- are they useless?
\medskip\noindent
At first sight, the answer seems to be positive, but classical results of
V. Strassen [20] and J. Morgenstern [13] tell us that the higher order
derivatives are not expensive to calculate, and can be computed
automatically. The purpose of this paper is to give an answer to the third
question by proving that the higher order derivatives of a function can be
computed with the same precision as the function itself.
We prove also that the derivatives so computed are
equal to the derivatives of the discrete problem (see Diagram 1). We
call the discrete
problem the finite dimensional problem processed by the computer. This result
 allows the use of automatic differentiation ([5], [6]), which works only on
discrete problems.
Furthermore, the computations of Taylor's expansions
which are proposed at the end of this paper, could be a partial answer to
the last question.

Mohamed Masmoudi

Papers

The Topological Asymptotic for PDE Systems: The Elasticity Case

Crack detection by the topological gradient method

The Topological Asymptotic for the Helmholtz Equation

The topological asymptotic expansion for the Maxwell equations and some applications

Computation of high order derivatives in optimal shape design