In the context of nuclear reactor safety, several passive systems have been studied, including the smart accumulator and the flow limiter. With the aim of optimising the efficiency of these devices, a geometrical optimization process using the penalized direct forcing method has been proposed. It is also possible to extend this optimization approach to the study of nuclear fuel shapes according to the material disposition. The particularity of this method is to describe the immersed boundary with a data field of barycenters, areas and normals. To achieve this, we can obtain the shape derivative of the objective function with the introduction of a Lagrangian. This approach is validated on a thermal test case with different initial shapes. A topology-conserving algorithm has been proposed to prevent topology breakage.

In this work, we propose and compare three numerical methods to handle the one-phase Hele-Shaw problem with surface tension in dimension two by using three variational approaches in the spirit of the seminal works \cite{Otto, Gia_Otto}.

SummaryThis article is concerned with the coupled optimization of the external boundary of a structure and its infill made of some graded lattice material. The lattice material is made of a periodic cell, macroscopically modulated and oriented. The external boundary may be coated by a layer of pure material with a fixed prescribed thickness. The infill is optimized by the homogenization method while the macroscopic shape is geometrically optimized by the Hadamard method of shape sensitivity. A first original feature of the proposed approach is that the infill material follows the displacement on the exterior boundary during the geometric optimization step. A second key feature is the dehomogenization or projection step which build a smoothly varying lattice infill from the optimal homogenized properties. Several numerical examples illustrate the effectiveness of our approach in 2‐d, which is especially convenient when considering design‐dependent loads.

Objective Intracranial electrical stimulation of subcortical axonal tracts is particularly useful during brain surgery, where mapping helps identify and excise dysfunctional tissue while avoiding damage to functional structures. Stimulation parameters are generally set empirically and consequences for the spatial recruitment of axons within subcortical tracts are not well identified.Approach Computational modeling is employed to study the effects of stimulation parameters on the recruitment of axons: monophasic versus biphasic stimuli induced with monopolar versus bipolar electrodes, oriented orthogonal or parallel to the tract, for isotropic and anisotropic tracts.Main results The area and depth of axonal activation strongly depend on tissue conductivity and electrode parameters. The largest activation area results from biphasic stimulation with bipolar electrodes oriented orthogonal to axonal fasciculi, for anisotropic and especially isotropic tracts. For anisotropic tracts, the maximal activation depth is similar regardless of whether a monopolar or bipolar electrode is employed. For isotropic tracts, bipolar parallel and monopolar stimulation activate axons deeper than orthogonal bipolar stimulation. Attention is warranted during monophasic stimulation: a blockade of action potentials immediately under cathodes and a propagation of action potentials under anodes are found.Significance Considering the spatial patterns of blockade and activation present during monophasic stimulation with both monopolar and bipolar electrodes, biphasic stimulation is recommended to explore subcortical axon responses during intraoperative mapping. Finally, the precise effect of electrical stimulation depends on conductivity profiles of tracts, and as such, should be explicitly considered for each individual subject and tract undergoing intracranial mapping.

This paper is concerned with the topology optimization of structures made of periodically perforated material, where the microscopic periodic cell can be macroscopically modulated and oriented. The main idea is to optimize the homogenized formulation of this problem, which is an easy task of parametric optimization, then to project the optimal microstructure at a desired lengthscale, which is a delicate issue, albeit computationally cheap. The main novelty of our work is, in a plane setting, the conformal treatment of the optimal orientation of the microstructure. In other words, although the periodicity cell has varying parameters and orientation throughout the computational domain, the angles between its members or bars are conserved. The main application of our work is the optimization of so-called lattice materials which are becoming increasingly popular in the context of additive manufacturing. Several numerical examples are presented for compliance minimization in 2-d.

The aim of this paper is to develop some techniques for automation of the mappings (between working and reference domains) required by reduced basis methods: the development of geometry mappings is indeed often a substantial impediment to the implementation of reduced basis techniques, especially in the context of the reduced basis element method (RBEM) and the reduced basis component method (RBCM). In the RBCM context, the geometry mappings are applied at the level of components. The methods have been tested on various cases to understand the limits of the approach and try to foresee and overcome the possible failures.

In this article, we consider the problem of optimal design of a compliant structure under a volume constraint, within the framework of linear elasticity. We introduce the pure displacement and the dual mixed formulations of the linear elasticity problem and we compute the volumetric expressions of the shape gradient of the compliance by means of the velocity method. A preliminary qualitative comparison of the two expressions of the shape gradient is performed through some numerical simulations using the Boundary Variation Algorithm.

We present here a topology optimization method based on a homogenization approach to design oriented and parametrized cellular structures. The present work deals with 2-D square cells featuring a rectangular hole, because their structure is close to that of rank-2 sequential laminates, which are optimal for compliance optimization. For several cells, the value and the parametric sensitivities of their effective elastic tensor can easily be computed, by the resolution of a cell problem. The obtained results can be used to build a surrogate model for the homogenized constitutive law. Moreover, we add the local orientation of the cells to our problem. Then, an optimal composite shape is computed thanks to an alternate directions algorithm. The crucial ingredient of the methodology is the extraction of a quasi-periodic and additive manufacturable structure from the previously obtained composite shape, based on the introduction of a space transformation.

In this paper we introduce a novel certified shape optimization strategy - named Certified Descent Algorithm (CDA) - to account for the numerical error introduced by the Finite Element approximation of the shape gradient. We present a goal-oriented procedure to derive a certified upper bound of the error in the shape gradient and we construct a fully-computable, constant-free a posteriori error estimator inspired by the complementary energy principle. The resulting CDA is able to identify a genuine descent direction at each iteration and features a reliable stopping criterion. After validating the error estimator, some numerical simulations of the resulting certified shape optimization strategy are presented for the well-known inverse identification problem of Electrical Impedance Tomography.

We consider a brittle elastic solid (prone to develop fractures) as the limit of a damage model and propose a numerical method to determine its quasi-static evolution in the spirit of Francfort and Marigo [3] and Allaire et al. [1,2]. (C) 2016 Academie des sciences. Publie par Elsevier Masson SAS.

The standard geometrical shape optimization method proceed by the application of successive diffeomorphisms close to the identity starting from an initial guessed shape. Consequently, it does not allow for the optimization of the topology which is kept unchanged from one iteration to the other. The topological gradient enables to determine if the inclusion of a small hole of given shape is cost efficient. Such holes can be included at any time during the geometrical shape optimization process. Moreover, as seen thereafter, the topological gradient can be explicitly computed from the primal and adjoint states of the optimization problem that are already computed during standard geometrical shape optimization. Finally let us mention that level set methods can also handle topological changes during the optimization process [2], possibly coupled with the use of topological gradient [1].

Biological membranes are often idealized as incompressible elastic surfaces whose strain energy only depends on their mean curvature and possibly on their shear. We show that this type of model can be derived using a formal asymptotic method by considering biological membranes to be thin, strongly anisotropic, elastic, locally homogeneous bodies.

In this paper we account for the numerical error introduced by the Finite Element approximation of the shape gradient to construct a guaranteed shape optimization method. We present a goal-oriented strategy inspired by the complementary energy principle to construct a constant-free, fully-computable a posteriori error estimator and to derive a certified upper bound of the error in the shape gradient. The resulting Adaptive Boundary Variation Algorithm (ABVA) is able to identify a genuine descent direction at each iteration and features a reliable stopping criterion for the optimization loop. Some preliminary numerical results for the inverse identification problem of Electrical Impedance Tomography are presented.

We derive the equivalent energy of a square lattice that either deforms into the three- dimensional Euclidean space or remains planar. Interactions are not restricted to pairs of points and take into account changes of angles. Under some relationships between the local energies associated with the four vertices of an elementary square, we show that the limit energy can be obtained by mere quasiconvexification of the elementary cell energy and that the limit process does not involve any relaxation at the atomic scale. In this case, it can be said that the Cauchy–Born rule holds true. Our results apply to classical models of mechanical trusses that include torques between adjacent bars and to atomistic models.

We address the problem of the numerical treatment of frictionless contacts and selfcontacts between nonlinear elastic bodies, moving in R 2 , in the presence of large deformations. We propose a new penalization approach, based on a modelling we have introduced in former works, to take into account the noninterpenetration constraint.

Traditional designs of sodium cooled fast reactors have a positive sodium expansion feedback. During a loss of flow transient without scram, sodium heating and boiling thus insert a positive reactivity and prevents the power from decreasing. Recent studies led at CEA, AREVA and EDF show that cores with complex geometries can feature a very low or even a negative sodium void worth.(1, 2) Usual optimization methods for core conception are based on a parametric description of a given core design(3).(4) New core concepts and shapes can then only be found by hand. Shape optimization methods have proven very efficient in the conception of optimal structures under thermal or mechanical constraints.(5, 6) First studies show that these methods could be applied to sodium cooled core conception.(7) In this paper, a shape optimization method is applied to the conception of a sodium cooled fast reactor core with low sodium void worth. An objective function to be minimized is defined. It includes the reactivity change induced by a 1% sodium density decrease. The optimization variable is a displacement field changing the core geometry from one shape to another. Additionally, a parametric optimization of the plutonium content distribution of the core is made, so as to ensure that the core is kept critical, and that the power shape is flat enough. The final shape obtained must then be adjusted to a get realistic core layout. Its caracteristics can be checked with reference neutronic codes such as ERANOS. Thanks to this method, new shapes of reactor cores could be inferred, and lead to new design ideas.