Research

Research Interests

I am broadly interested in the concrete applications of mathematics, particularly in industrially motivated problems that involve meaningful connections with shape optimization, inverse problems, numerical analysis, and other related areas.

My current work focuses on data completion problems, applications of sweeping processes, and aspects of infinite-dimensional affine eigenvalue problems. I also have a strong interest—still at an exploratory stage—in applying machine learning to shape optimization, mechanics, and beyond, as well as in stochastic simulation techniques for financial applications.

Below are the essentials (abstracts+some images) of my publications.

History-depending Sweeping Processes

In this paper, we study the well-posedness of state-dependent and state-independent sweeping processes driven by prox-regular sets and perturbed by a history-dependent operator. Our approach, based on an enhanced version of Gronwall’s lemma and fixed-point arguments, provides an efficient framework for analyzing sweeping processes. In particular, our findings recover all existing results for the class of Volterra sweeping processes and provide new insights into history-dependent sweeping processes. Finally, we apply our theoretical results to establish the well-posedness of a viscoelastic model with long memory.

Keywords: Sweeping process, Fixed point theory, History-dependent operator, prox-regular set.

Support Optimization: Accessibility

This paper is concerned with a geometric constraint, the so-called accessibility constraint, for shape and topology optimization of structures built by additive manufacturing. The motivation comes from the use of sacrificial supports to maintain a structure, submitted to intense thermal residual stresses during its building process. Once the building stage is finished, the supports are no longer useful and should be removed. However, such a removal can be very difficult or even impossible if the supports are hidden deep inside the complex geometry of the structure. A rule of thumb for evaluating the ease of support removal is to ask that the contact zone between the structure and its supports can be accessed from the exterior by a straight line which does not cross another part of the structure. It mimicks the possibility to cut the head of the supports attached to the structure with some cutting tool. The present work gives a new mathematical way to evaluate such an accessibility constraint, which is based on distance functions, solutions of eikonal equations. The main advantage is the possibility of computing shape derivatives of such a criterion with respect to both the structure and the support. We numerically demonstrate in 2D and 3D that, in the context of the level-set method for topology optimization, such an approach allows us to optimize simultaneously the mechanical performance of a structure and the accessibility of its building supports, guaranteeing its manufacturability.

Keywords: Topology optimization, Additive manufacturing, Support, Accessibility, Level set.

Support Optimization: Shape Optimization on Additive Manufacturing

Supports are an important ingredient of the building process of structures by additive manufacturing technologies. They are used to reinforce overhanging regions of the desired structure and/or to facilitate the mitigation of residual thermal stresses due to the extreme heat flux produced by the source term (laser beam). Very often, supports are, on purpose, weakly connected to the built structure for easing their removal.

In this work we consider an imperfect interface model for which the interaction between supports and the built structure is not ideal, meaning that the displacement is discontinuous at the interface while the normal stress is continuous and proportional to the jump of the displacement. The optimization process is based on the level set method, body fitted meshes and the notion of shape derivative using the adjoint method. Completely different designs of supports are obtained with perfect or imperfect interfaces.

Keywords: Additive manufacturing, Level set method, Imperfect interface, Imperfect bonding, Tool Accessibility, Support optimization, Hadamard method, Shape optimization, Shape gradient, Overhang, Anisotropic supports.

Caption: In order to build a table (upper left) subject to its own weight supports are needed. On the lower left we can see the obtained distribution of supports for a perfect interface model for the support-piece interface such that the compliance is minimized (i.e. providing the most rigid piece+support structure), on the right (upper and lower) the distribution of supports for an (very) imperfect interface: supports are distributed even over the table, as the imperfect interaction may cause a total decoupling for the lower supports, and the upper ones provide additional rigidity.

Shape Optimization for Imperfect Interfaces

In the context of a diffusion equation, this work is devoted to a two-phase optimal design problem where the interface, separating the phases, is imperfect, meaning that the solution is discontinuous while the normal flux is continuous and proportional to the jump of the solution. The shape derivative of an objective function with respect to the interface position is computed by the adjoint method. Numerical experiments are performed with the level set method and an exact remeshing algorithm so that the interface is captured by the mesh at each optimization iteration.

Keywords: Imperfect interface, Imperfect bonding, Hadamard boundary variation method, Shape differentiability, Shape gradient, Level set method.

Caption: Interface optimization results for perfect interface (center) and imperfect interface (right). Problem setting is depicted on the left.

Previous Research Topics

Parameter identification in tumoral modelling

In this work we have developped a general mathematical model and devise a practical identifiability approach for gastrointestinal stromal tumor (GIST) metastasis to the liver, with the aim of quantitatively describing therapy failure due to drug resistance. To this end, we have modeled metastatic growth and therapy failure produced by resistance to two standard treatments that have been observed clinically in patients with GIST metastasis to the liver. We propose a general modeling framework based on ODE for GIST metastatic growth and therapy failure due to drug resistance and analyzed five different model variants, using medical image observations (CT scans) from patients that exhibit drug resistance. The associated parameter estimation problem was solved using the Nelder-Mead simplex algorithm, by adding a regularization term to the objective function to address model instability, and assessing the agreement of either an absolute or proportional error in the objective function.

Keywords: Tikhonov regularization, GIST, Nelder-Mead simplex algorithm, Parameter identification.

Obstacle detection and data completion problems

We have worked in the study of the inverse problem of obstacle/object detection using optimization methods. This problem consists in localizing an unknown object ω inside a known bounded domain Ω by means of boundary measurements and more precisely by a given Cauchy pair on a part Γobs of ∂Ω.

The strategy used in this work is to reduce the inverse problem into the minimization of a cost-type functional: the Kohn-Vogelius functional. This kind of approach is widely used and permits to use optimization tools for numerical implementations. However, in order to well-define the functional, this approach needs to assume the knowledge of a measurement on the whole exterior boundary ∂Ω.

This last point leads to the study of the so-called the data completion problem which consists in recovering the boundary conditions on an inaccessible region, i.e. on ∂Ω\Γobs , from the Cauchy data on the accessible region Γobs . This inverse problem is also studied through the minimization of a Kohn-Vogelius type functional. The ill-posedness of this problem enforces us to regularize the functional via a Tikhonov regularization.

In order to perform the numerical reconstruction of the unknown inclusion, we rely on shape optimization methods: Topological gradient methods in order to determine the number of objects and Hadamard's boundary variation method in order to reconstruct the desired shape.

Keywords: Geometrical inverse problem, Shape optimization, Data completion problem, Topological sensitivity analysis, Topological gradient, Shape gradient, Kohn-Vogelius functional.

Picture: Camino al Volcán (km ~70), San José de Maipo, Chile.

Last update (this section): April 1st, 2022.

Page updated

Google Sites

Report abuse