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In recent years, the field of additive manufacturing (AM), often referred to as 3D printing, has seen tremendous growth and radically changed the means by which we describe valid 3D models for production. In particular, it is now conceivable to produce composite structures consisting of smoothly varying oriented anisotropic constitutive materials. In the present work, we propose a sensitivity driven method for the generation of transverse isotropic fiber reinforced structures having smooth spatially varying orientations. Our approach builds upon finite element analysis (FEA) and density-based topology optimization (TO). The local material orientations are formulated as design variables in a stiffness maximization problem, and solved with a non-convex gradient-based optimization scheme. Length-scale control is achieved through the use of filters for regularization. We demonstrate the ability of the proposed approach to handle large-scale 3D problems with synchronous optimization of material densities and orientations yielding millions of design variables on multiple load case scenarios. The method is shown to be compatible with compliant mechanism optimization as well as local volume constraints. Finally, the approach is extended with an additional design variable dictating the ratio of anisotropy for each element, thereby delegating the choice of material type to the optimization scheme.

International audience; The numerical simulation of time-harmonic waves in heterogeneous media is a tricky task which consists in reproducing oscillations. These oscillations become stronger as the frequency increases, and high-order finite element methods have demonstrated their capability to reproduce the oscilla-tory behavior. However, they keep coping with limitations in capturing fine scale heterogeneities. We propose a new approach which can be applied in highly heterogeneous propagation media. It consists in constructing an approximate medium in which we can perform computations for a large variety of frequencies. The construction of the approximate medium can be understood as applying a quadrature formula locally. We establish estimates which generalize existing estimates formerly obtained for homogeneous Helmholtz problems. We then provide numerical results which illustrate the good level of accuracy of our solution methodology.

High order finite element methods are very efficient to approximate the solution of wave propagation problems in the high frequency regime when the propagation medium is homogeneous. Unfortunately, theses methods fail to handle small-scale heterogeneities if the mesh is not properly constrained. In geophysical applications, constraining the mesh to the size of small-scale heterogeneities might be unaffordable so that high order finite element methods can not be applied directly. We propose to overcome this limitation by considering a multiscale strategy, which makes it possible to take into account small-scale heterogeneities on coarse meshes. Our multiscale procedure is equivalent to a quadrature-like formula. It thus corresponds to a pre-processing step in the computations which can be easily parallelized. Hence, the overhead of our multiscale technique is negligible. We validate the accuracy and the efficiency of our methodology on 2D and 3D geophysical benchmarks.

International audience; In many problems of geophysical interest, one has to deal with data that exhibit complex fault structures. This occurs, for instance, when describing the topography of seafloor surfaces, mountain ranges, volcanoes, islands, or the shape of geological entities, as well as when dealing with reservoir characterization and modelling. In all these circumstances, due to the presence of large and rapid variations in the data, attempting a fitting using conventional approximation methods necessarily leads to instability phenomena or undesirable oscillations which can locally and even globally hinder the approximation. As will be shown in this paper, the right approach to get a good approximant consists, in effect, in applying first a segmentation process to precisely define the locations of large variations and faults, and exploiting then a discrete approximation technique. To perform the segmentation step, we propose a quasi-automatic algorithm that uses a level set method to obtain from the given (gridded or scattered) Lagrange data several patches delimited by large gradients (or faults). Then, with the knowledge of the location of the discontinuities of the surface, we generate a triangular mesh (which takes into account the identified set of discontinuities) on which a $D^m$ -spline approximant is constructed. To show the efficiency of this technique, we will present the results obtained by its application to synthetic datasets as well as real gridded datasets in Oceanography and Geosciences.

We present a deformable model technique for geophysical image analysis. Deformable model approaches have been developed extensively in the literature, including prior applications to geophysical or medical image interpretation. In this paper we propose a method to segment a geophysical image under interpolation conditions (well data). The originality of this segmentation method is that it considers the deformable model as a set of articulated curves, which corresponds to the interfaces between different regions. Moreover, the interpolation conditions permit some geometric constraints to be made on the model. The theoretical aspect of the method is given in the case of a three dimensional image. Numerical results are given.

We introduce a surface approximation technique to address the problem of fitting a surface to a given set of curves. The originality of the method lies in its ability to take into account the continuous aspect of the data, and also in the possibility to arbitrarily select the regularity (C0, C1, or higher) of the approximant obtained. We demonstrate the efficiency of the approach by constructing a bathymetry map of the Marianas trench based upon a set of SONAR (SOnic Navigation And Ranging) bathymetry ship track data.

Image inpainting consists in reconstructing missing parts of a given image. In this work, we propose to use an approximation method based on splines and finite elements approximation to recover missing depth values in images acquired with a Kinect 3D sensor. Neighboring pixels in the depth map may contain very different distance values. The considered surface approximation problem therefore involves rapidly varying data which can lead to oscillations (Gibbs phenomenon). To address this issue, we propose to apply two scale transformations to dampen these oscillations near steep gradients implied by the data. The algorithm is presented with some numerical examples of inpainting. Our approach also permits to get a finer resolution of the 3D depth map.

Scale transformations are common in approximation. In surface approximation from rapidly varying data, one wants to suppress, or at least dampen the oscillations of the approximation near steep gradients implied by the data. In that case, scale transformations can be used to give some control over overshoot when the surface has large variations of its gradient. Conversely, in image analysis, scale transformations are used in preprocessing to enhance some features present on the image or to increase jumps of grey levels before segmentation of the image. In this paper, we establish the convergence of an approximation method which allows some control over the behavior of the approximation. More precisely, we study the convergence of an approximation from a data set \(\{ x_i ,f(x_i )\} \)of \(\mathbb{R}^n \times \mathbb{R} \), while using scale transformations on the \(f(x_i ) \)values before and after classical approximation. In addition, the construction of scale transformations is also given. The algorithm is presented with some numerical examples.

In this paper, we study the problem of constructing a smooth approximant of a surface defined by the equation z = f(x1, x2), the data being a finite set of patches on this surface. This problem occurs, for example, after geophysical processing such as migration of time-maps or depth-maps. The usual algorithms to solve this problem are picking points on the patches to get Lagrange's data or trying to get local junctions on patches. But the first method does not use the continuous aspect of the data and the second one does not perform well to get a global regular approximant (C1 or more). As an approximant of f, a discrete smoothing spline belonging to a suitable piecewise polynomial space is proposed. The originality of the method consists in the fidelity criterion used to fit the data, which takes into account their particular aspect (surface's patches): the idea is to define a function that minimizes the volume located between the data patches and the function, and which is globally Ck. We first demonstrate the new method on a theoretical aspect and numerical results on real data are given.

A collocation method based on radial basis functions (RBF) is introduced for wind field adjustment in meteorology. The numerical solutions are shown to be more accurate than those obtained with the finite element method (FEM), and they also require much less computational effort. A detailed analysis shows how inconsistent boundary conditions may affect numerical solutions.

cited By (since 1996)0; International audience; In this paper, we investigate a new method to enforce topology preservation on two/three-dimensional deformation fields for non-parametric registration problems involving large-magnitude deformations. The method is composed of two steps. The first one consists in correcting the gradient vector field of the deformation at the discrete level, in order to fulfill a set of conditions ensuring topology preservation in the continuous domain after bilinear interpolation. This part, although related to prior works by Karacali and Davatzikos (Estimating Topology Preserving and Smooth Displacement Fields, B. Karacali and C. Davatzikos, IEEE Transactions on Medical Imaging, vol. 23(7), 2004), proposes a new approach based on interval analysis and provides, unlike their method, uniqueness of the correction parameter α at each node of the grid, which is more consistent with the continuous setting. The second one aims to reconstruct the deformation, given its full set of discrete gradient vector field. The problem is phrased as a functional minimization problem on a convex subset K of an Hilbert space V. Existence and uniqueness of the solution of the problem are established, and the use of Lagrange's multipliers allows to obtain the variational formulation of the problem on the Hilbert space V. The discretization of the problem by the finite element method does not require the use of numerical schemes to approximate the partial derivatives of the deformation components and leads to solve two/three uncoupled sparse linear subsystems. Experimental results in brain mapping and comparisons with existing methods demonstrate the efficiency and the competitiveness of the method. © 2012 SPIE.

cited By (since 1996)1; International audience; In this paper, we investigate a new method to enforce topology preservation on deformation fields. The method is composed of two steps. The first one consists in correcting the gradient vector fields of the deformation at the discrete level, in order to fulfill a set of conditions ensuring topology preservation in the continuous domain after bilinear interpolation. This part, although related to prior works by Karaçali and Davatzikos, proposes a new approach based on interval analysis. The second one aims to reconstruct the deformation, given its full set of discrete gradient vectors. The problem is phrased as a functional minimization problem on the convex subset K of the Hilbert space V. The existence and uniqueness of the solution of the problem are established, and the use of Lagrange's multipliers allows to obtain the variational formulation of the problem on the Hilbert space V. Experimental results demonstrate the efficiency of the method. © 2011 IEEE.

In this paper we propose a segmentation method to locate faults and/or large variations in order to approximate a surface from Lagrange dataset.

In this paper, we propose an approximation method for surfaces with fault and /or large variation. We use image segmentation tools, meshing constraints, finite element methods and spline theory.