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An approach for optimal control of the interface between water and ferrofluid in a 2-D two-phase flow is proposed in the presence of a magnetic field generated by a matrix of driving electromagnets. First, a model combining Navier-Stokes equations and level set methods is developed. Since it is very computationally demanding, an approximate black-box model based on neural networks replacing the original model is constructed for the purpose of control design. In particular, one-hidden-layer feedforward neural networks with a different number of neurons are trained to predict the water-ferrofluid behavior with accuracy. Then, optimal control based on such black-box models is addressed by selecting the currents flowing in the electromagnets that minimize a cost function given by the symmetric difference between the desired shape and the actual interface separating water and ferrofluid. Numerical results based on both simulation and experimental data collected on the field showcase the effectiveness of the proposed approach.

A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly defined geometry as the graph of an implicitly defined, multi-valued height function, and applies a dimension reduction approach needing only one-dimensional quadrature. In particular, we explore the use of Gauss-Legendre and tanh-sinh methods and demonstrate that the quadrature algorithm inherits their high-order convergence rates. Under the action of $h$-refinement with $q$ fixed, the quadrature schemes yield an order of accuracy of $2q$, where $q$ is the one-dimensional node count; numerical experiments demonstrate up to 22nd order. Under the action of $q$-refinement with the geometry fixed, the convergence is approximately exponential, i.e., doubling $q$ approximately doubles the number of accurate digits of the computed integral. Complex geometry is automatically handled by the algorithm, including, e.g., multi-component domains, tunnels, and junctions arising from multiple polynomial level sets, as well as self-intersections, cusps, and other kinds of singularities. A variety of numerical experiments demonstrates the quadrature algorithm on two- and three-dimensional problems, including: randomly generated geometry involving multiple high-curvature pieces; challenging examples involving high degree singularities such as cusps; adaptation to simplex constraint cells in addition to hyperrectangular constraint cells; and boolean operations to compute integrals on overlapping domains., 44 pages, 18 figures, 4 algorithms, 1 table

The final publication is available at Springer via http://dx.doi.org/10.1007/s10237-020-01291-x This work explores the use of an embedded computational fluid dynamics method to study the type B aortic dissection. The use of the proposed technique makes it possible to easily test different intimal flap configurations without any need of remeshing. To validate the presented methodology, we take as reference test case an in vitro experiment present in the literature. This experiment, which considers several intimal flap tear configurations (number, size and location), mimics the blood flow in a real type B aortic dissection. We prove the correctness and suitability of the presented approach by comparing the pressure values and waveform. The obtained results exhibit a remarkable similarity with the experimental reference data. Complementary, we present a feasible surgical application of the presented computer method. The aim is to help the clinicians in the decision making before the type B aortic dissection surgical fenestration. The capabilities of the proposed technique are exploited to efficiently create artificial reentry tear configurations. We highlight that only the radius and center of the reentry tear need to be specified by the clinicians, without any need to modify neither the model geometry nor the mesh. The obtained computational surgical fenestration results are in line with the medical observations in similar clinical studies.

In the formulation of shape optimization problems, multiple geometric constraint functionals involve the signed distance function to the optimized shape Ω. The numerical evaluation of their shape derivatives requires to integrate some quantities along the normal rays to Ω, a challenging operation to implement, which is usually achieved thanks to the method of characteristics. The goal of the present paper is to propose an alternative, variational approach for this purpose. Our method amounts, in full generality, to compute integral quantities along the characteristic curves of a given velocity field without requiring the explicit knowledge of these curves on the spatial discretization; it rather relies on a variational problem which can be solved conveniently by the finite element method. The well-posedness of this problem is established thanks to a detailed analysis of weighted graph spaces of the advection operator β ⋅ ∇ associated to a C1 velocity field β. One novelty of our approach is the ability to handle velocity fields with possibly unbounded divergence: we do not assume div(β) ∈ L∞. Our working assumptions are fulfilled in the context of shape optimization of C2 domains Ω, where the velocity field β = ∇dΩ is an extension of the unit outward normal vector to the optimized shape. The efficiency of our variational method with respect to the direct integration of numerical quantities along rays is evaluated on several numerical examples. Classical albeit important implementation issues such as the calculation of a shape’s curvature and the detection of its skeleton are discussed. Finally, we demonstrate the convenience and potential of our method when it comes to enforcing maximum and minimum thickness constraints in structural shape optimization.

The focus of this article is on level set based topology optimization of vibroacoustic hearing instruments. The goal is to demonstrate the applicability of the proposed framework for optimization of 3D industrial scale hearing instruments. The framework employs an immersed boundary cut element method to handle the modeling of complex design geometries on fixed unstructured meshes. Utilization of unstructured meshes allows for optimizing small parts of a complex hearing instrument without disturbing the overall geometry. The remaining parts of the model which are not included in the optimization process are modeled with segregated approach. The design parameterization is based on an explicit level set approach, for which the nodal level set values are linked to the mathematical design variables. The optimization problem is solved using mathematical programming and the sensitivities are obtained with a discrete adjoint approach. A validation study is carried out comparing the proposed cut element model to a body fitted mesh for a large range of frequencies. The optimization framework is then applied on the tube and the suspension structures of a hearing instrument system considering two sets of material properties for the design parts. The optimization considers the minimization of sound pressure on the microphone surface for discrete frequencies aiming to reduce the feedback paths and to increase the amplification that the device can deliver to the user. Both optimization cases improve the performance of the hearing instrument system by effectively reducing the sound pressure on the microphone surface for the considered frequency range.

We introduce a deep learning framework designed to train smoothed elastoplasticity models with interpretable components, such as the stored elastic energy function, yield surface, and plastic flow that evolve based on a set of deep neural network predictions. By recasting the yield function as an evolving level set, we introduce a deep learning approach to deduce the solutions of the Hamilton–Jacobi equation that governs the hardening/softening mechanism. This machine learning hardening law may recover any classical hand-crafted hardening rules and discover new mechanisms that are either unbeknownst or difficult to express with mathematical expressions. Leveraging Sobolev training to gain control over the derivatives of the learned functions, the resultant machine learning elastoplasticity models are thermodynamically consistent, interpretable, while exhibiting excellent learning capacity. Using a 3D FFT solver to create a polycrystal database, numerical experiments are conducted and the implementations of each component of the models are individually verified. Our numerical experiments reveal that this new approach provides more robust and accurate forward predictions of cyclic stress paths than those obtained from black-box deep neural network models such as the recurrent neural network, the 1D convolutional neural network, and the multi-step feed-forward model.

We present a parallel method for solving the eikonal equation associated with level set redistancing. Fast marching [1] , [2] and fast sweeping [3] are the most popular redistancing methods due to their efficiency and relative simplicity. However, these methods require propagation of information from the zero-isocontour outwards, and this data dependence complicates efficient implementation on today's multiprocessor hardware. Recently an interesting alternative view has been developed that utilizes the Hopf–Lax formulation of the solution to the eikonal equation [4] , [5] . In this approach, the signed distance at an arbitrary point is obtained without the need of distance information from neighboring points. We extend the work of Lee et al. [4] to redistance functions defined via interpolation over a regular grid. The grid-based definition is essential for practical application in level set methods. We demonstrate the effectiveness of our approach with GPU parallelism on a number of representative examples.

The dynamics of binary droplets collision is of huge importance in different fields, from multiphase reactors (Dudukovic et al in Chem Eng Sci 54:1975–1995, 1999 [11]), raindrop formation, ink-jet printing, spray combustion, emulsion stability, turbine blade cooling, spray coating (Ashgriz and Poo in J Fluid Mech 221:183–204, 1990 [5]), and to drug delivery. Due to the complexity of the nature of droplets collision, this topic is one of the most challenging areas in the field of fluid dynamics. We acknowledge Ministerio de Economía y Competitividad, Secretaría de Estado de Investigación, Desarrollo e Innovación, Spain (ENE2017-88697-R). A. Amani acknowledges the financial support of (AGAUR) of Generalitat de Catalunya research scholarship (2016 FI_B 01059). N. Balcázar acknowledges the Programa Torres Quevedo MINECO (PTQ-14-07186).

This work presents Direct Numerical Simulation of mass transfer from buoyancy-driven bubbles rising in a wall-confined vertical channel, through a multiple markers level-set method. The Navier-Stokes equations and mass transfer equation are discretized using a finite volume method on a collocated unstructured mesh, whereas a multiple markers approach is used to avoid the numerical coalescence of bubbles. This approach is based on a mass conservative level-set method. Furthermore, unstructured flux-limiter schemes are used to discretize the convective term of momentum equation, level-set advection equations, and mass transfer equation, to improve the stability of the solver in bubbly flows with high Reynolds number and high-density ratio. The level-set model is used to research the effect of bubble-bubble and bubble-wall interactions on the mass transfer from a bubble swarm rising in a vertical channel with a circular cross-section.

We address the optimal control of level sets associated with the solution of normal flow equations. The problem consists in finding the normal velocity to the front described by a certain level set in such a way to minimize a given cost functional. First, the considered problem is shown to admit a solution on a suitable space of functions. Then, since in general it is difficult to solve it analytically, an approximation scheme that relies on the extended Ritz method is proposed to find suboptimal solutions. Specifically, the control law is forced to take on a neural structure depending nonlinearly on a finite number of parameters to be tuned, i.e., the neural weights. The selection of the optimal weights is performed with two different approaches. The first one employs classical line-search descent methods, while the second one is based on a quasi-Newton optimization that can be regarded as a neural learning based on the extended Kalman filter. If compared to line-search methods, such an approach reveals to be successful with a reduced computational effort and an increased robustness with respect to the trapping into local minima, as confirmed by simulations in both two and three dimensions.

The behavior of rotating and/or charged drops is a classic problem in fluid mechanics with a multitude of industrial applications. Theoretical studies of such liquid drops have also provided important insights into fundamental physical processes across nuclear and astrophysical lengthscales. However, the full nonlinear dynamics of these drops are only just beginning to be uncovered by experiments. These nonlinear effects are manifest in the high sensitivity of the breakup mechanisms to small perturbations of the initial drop shape and in observations of hysteresis in the transition between different drop shape families. This paper investigates the equilibrium shapes and stability of charged and rotating drops in a vacuum with an energy minimization method applied to spheroidal shapes and with numerical simulations using a finite-difference, level-set method. A good working formula for the stability limit of these drops is given by Lmax = 1.15 − 0.59x − 0.56x2, where L is the dimensionless angular momentum and x is the charge fissility parameter. These methods also provide a firm explanation for the hysteresis of rotating and charged drops.

A novel multiple marker level-set method is introduced for Direct Numerical Simulation of deformable fluid particles (bubbles and droplets), which is integrated in a finite-volume framework on collocated unstructured grids. Each fluid particle is described by a separate level-set function, thus, different interfaces can be solved in the same control volume, avoiding artificial and potentially unphysical coalescence of fluid particles. Therefore, bubbles or droplets are able to approach each other closely, within the size of one grid cell, and can even collide. The proposed algorithm is developed in the context of the conservative level-set method, whereas, surface tension is modeled by the continuous surface force approach. The pressure-velocity coupling is solved by the fractional-step projection method. For validation of the proposed numerical method, the gravity-driven impact of a droplet on a liquid-liquid interface is studied; then, the binary droplet collision with bouncing outcome is examined, and finally, it is applied on simulation of gravity-driven bubbly flow in a vertical column.

The goal of this work is the optimal control of level sets generated by the normal flow equation. The problem consists in finding the normal velocity that minimizes a given performance index expressed by means of a cost functional. In this perspective, a sufficient condition of optimality requiring the solution of a system of partial differential equations is derived. As in general it is difficult to solve such a system, an approximation scheme based on the extended Ritz method is proposed to find suboptimal solutions. The control law is forced to take on a fixed structure depending nonlinearly on a finite number of parameters to be suitably chosen. The selection of the parameters is accomplished by using a gradient-based technique. To this end, the adjoint equation is derived to compute the gradient of the cost functional with respect to the parameters of the control law. The effectiveness of the proposed approach is shown by means of simulations.

The computation of multiphase flows presents a subtle energetic equilibrium between potential (i.e., surface) and kinetic energies. The use of traditional interface-capturing schemes provides no control over such a dynamic balance. In the spirit of the well-known symmetry-preserving and mimetic schemes, whose physics-compatible discretizations rely upon preserving the underlying mathematical structures of the space, we identify the corresponding structure and propose a new discretization strategy for curvature. The new scheme ensures conservation of mechanical energy (i.e., surface plus kinetic) up to temporal integration. Inviscid numerical simulations are performed to show the robustness of such a method.

Among the various nonlinear image filtering techniques, the Affine Morphological Scale Space (AMSS) model is characterized by especially appealing theoretical properties. In this paper we propose a Semi-Lagrangian scheme to approximate the AMSS model, prove convergence of its monotone version and apply it to some typical problem of image filtering. We also carry out a comparison with the Level Lines Affine Shortening implementation of the AMSS model, as well as with other nonlinear filtering techniques, namely the Mean Curvature equation and the Perona-Malik model.

CFD modeling of turbulent free surface flows has become an important tool in the design of ship hulls. A two-fluid flow solver that can predict the flow pattern, free surface shape and the forces on the ship hull is presented. The discretization is based on unstructured linear finite elements, tetrahedras and prisms. A Variational Multiscale Stabilization technique known as Algebraic Sub Grid Scale Stabilization (ASGS) is used to deal with convection dominated flows and allow for equal order velocity–pressure interpolations. A fixed grid method that captures the position of the interface with the Level Set technique is used to simulate the two-phase flow. The jump in the fluid properties is smoothed in a region close to the interface. Spalart Allmaras and SST k–ω turbulence models have been tested without showing significant differences. The need to accurately predict the viscous forces on the ship hull has motivated the use of anisotropically refined prism elements close the hull. Such meshes have challenged the capabilities of the equation based reinitialization techniques that we had previously used for the Level Set method.

We present a shape prior constraint to guide the evolution of implicit active contours. Our method includes three core techniques. Firstly, a rigid registration is introduced, using a line search method within a level set framework. The method automatically finds the time step for the iterative optimization processes. The order for finding the optimal translation, rotation and scale is derived experimentally. Secondly, a single reconstructed shape is created from a shape distribution of a previously acquired learning set. The reconstructed shape is applied to guide the active contour evolution. Thirdly, our method balances the impact of the shape prior versus the image guidance of the active contour. A mixed stopping condition is defined based on the stationarity of the evolving curve and the shape prior constraint. Our method is completely non-parametric and avoids taking linear combinations of non-linear signed distance functions, which would cause problems because distance functions are not closed under linear operations. Experimental results show that our method is able to extract the desired objects in several circumstances, namely when noise is present in the image, when the objects are in slightly different poses and when parts of the object are invisible in the image.

We propose a storage efficient, fast and parallelizable out-of-core framework for streaming computations of high resolution level sets. The fundamental techniques are skewing and tiling transformations of streamed level set computations which allow for the combination of interface propagation, re-normalization and narrow-band rebuild into a single pass over the data stored on disk. When combined with a new data layout on disk, this improves the overall performance when compared to previous streaming level set frameworks that require multiple passes over the data for each time-step. As a result, streaming level set computations are now CPU bound and consequently the overall performance is unaffected by disk latency and bandwidth limitations. We demonstrate this with several benchmark tests that show sustained out-of-core throughputs close to that of in-core level set simulations.

We present an overview of various edge and line oriented approaches to contour detection that have been proposed in the last two decades. By edge and line oriented we mean methods that do not rely on segmentation. Distinction is made between edges and contours. Contour detectors are divided in local and global operators. The former are mainly based on differential analysis, statistical approaches, phase congruency, rank order filters, and combinations thereof. The latter include computation of contour saliency, perceptual grouping, relaxation labeling and active contours. Important aspects are covered, such as preprocessing aimed to suppress texture and noise, multiresolution techniques, connections between computational models and properties of the human visual system, and use of shape priors. An overview of procedures and metrics for quantitative performance evaluation is also presented. Our main conclusion is that contour detection has reached high degree of sophistication, taking into account multimodal contour definition (by luminance, color or texture changes), mechanisms for reducing the contour masking influence of noise and texture, perceptual grouping, multiscale aspects and high-level vision information. (C) 2010 Elsevier B.V. All rights reserved.

The aim of this work is to introduce and design image processing methods for the quantitative analysis of epicardial fat by using cardiac CT imaging. Indeed, epicardial fat has recently been shown to correlate with cardiovascular disease, cardiovascular risk factors and metabolic syndrome. However, many concerns still remain about the methods for measuring epicardial fat, its regional distribution on the myocardium and the accuracy and reproducibility of the measurements. In this paper, a method is proposed for the analysis of single-frame 3D images obtained by the standard acquisition protocol used for coronary calcium scoring. In the design of the method, much attention has been payed to the minimization of user intervention and to reproducibility issues. In particular, the proposed method features a two step segmentation algorithm suitable for the analysis of epicardial fat. In the first step of the algorithm, an analysis of epicardial fat intensity distribution is carried out in order to define suitable thresholds for a first rough segmentation. In the second step, a variational formulation of level set methods - including a specially-designed region homogeneity energy based on Gaussian mixture models- is used to recover spatial coherence and smoothness of fat depots. Experimental results show that the introduced method may be efficiently used for the quantification of epicardial fat.

International audience; In this paper, we propose a segmentation method based on the generalized fast marching method (GFMM) developed by Carlini et al. (submitted). The classical fast marching method (FMM) is a very efficient method for front evolution problems with normal velocity (see also Epstein and Gage, The curve shortening flow. In: Chorin, A., Majda, A. (eds.) Wave Motion: Theory, Modelling and Computation, 1997) of constant sign. The GFMM is an extension of the FMM and removes this sign constraint by authorizing time-dependent velocity with no restriction on the sign. In our modelling, the velocity is borrowed from the Chan-Vese model for segmentation (Chan and Vese, IEEE Trans Image Process 10(2):266-277, 2001). The algorithm is presented and analyzed and some numerical experiments are given, showing in particular that the constraints in the initialization stage can be weakened and that the GFMM offers a powerful and computationally efficient algorithm.

We propose a Semi-Lagrangian scheme coupled with Radial Basis Function interpolation for approximating a curvature-related level set model, which has been proposed by Zhao et al. in \cite{ZOMK} to reconstruct unknown surfaces from sparse, possibly noisy data sets. The main advantages of the proposed scheme are the possibility to solve the level set method on unstructured grids, as well as to concentrate the reconstruction points in the neighbourhood of the data set, with a consequent reduction of the computational effort. Moreover, the scheme is explicit. Numerical tests show the accuracy and robustness of our approach to reconstruct curves and surfaces from relatively sparse data sets., Comment: 14 pages, 26 figures

A new method for segmenting white blood cells nuclei in microscopic images is presented. Challenges to accurate segmentation include intra-class variation of the nuclei cell boundaries, non-uniform illumination, and changes in the cell topology due to its orientation and stage of maturity. In this research, level set methods and geometric active contours are used to segment the nucleus of white blood cells from the cytoplasm and the cell wall. Level set methods use morphological operations to estimate an initial cell boundary and are fully automated. Geometric active contours are less computationally complex and adapt better to the curve topology of the cell boundary than parametric active contours, which have been previously used for white blood cell segmentation. Segmentation performance is compared with other segmentation methods using the Berkeley benchmark database and the proposed method is shown to be superior using various indices.

Data-extrapolating (extension) technique has important applications in image processing on implicit surfaces and in level set methods. The existing data-extrapolating techniques are inefficient because they are designed without concerning the specialities of the extrapolating equations. Besides, there exists little work on locating the narrow band after data extrapolating—a very important problem in narrow band level set methods. In this paper, we put forward the general Huygens’ principle, and based on the principle we present two efficient data-extrapolating algorithms. The algorithms can easily locate the narrow band in data extrapolating. Furthermore, we propose a prediction–correction version for the data-extrapolating algorithms and the corresponding band locating method for a special case where the direct band locating method is hard to apply. Experiments demonstrate the efficiency of our algorithms and the convenience of the band locating method.

© 2015 Society for Industrial and Applied Mathematics. A high-order accurate numerical quadrature algorithm is presented for the evaluation of integrals over curved surfaces and volumes which are defined implicitly via a fixed isosurface of a given function restricted to a given hyperrectangle. By converting the implicitly defined geometry into the graph of an implicitly defined height function, the approach leads to a recursive algorithm on the number of spatial dimensions which requires only one-dimensional root finding and one-dimensional Gaussian quadrature. The computed quadrature scheme yields strictly positive quadrature weights and inherits the high-order accuracy of Gaussian quadrature: a range of different convergence tests demonstrate orders of accuracy up to 20th order. Also presented is an application of the quadrature algorithm to a high-order embedded boundary discontinuous Galerkin method for solving partial differential equations on curved domains.

Segmentation of injured or unusual anatomic structures in medical imagery is a problem that has continued to elude fully automated solutions. In this paper, the goal of easy-to-use and consistent interactive segmentation is transformed into a control synthesis problem. A nominal level set partial differential equation (PDE) is assumed to be given; this open-loop system achieves correct segmentation under ideal conditions, but does not agree with a human expert's ideal boundary for real image data. Perturbing the state and dynamics of a level set PDE via the accumulated user input and an observer-like system leads to desirable closed-loop behavior. The input structure is designed such that a user can stabilize the boundary in some desired state without needing to understand any mathematical parameters. Effectiveness of the technique is illustrated with applications to the challenging segmentations of a patellar tendon in magnetic resonance and a shattered femur in computed tomography.

Nonlinear parametric inverse problems appear in several prominent applications; one such application is Diffuse Optical Tomography (DOT) in medical image reconstruction. Such inverse problems present huge computational challenges, mostly due to the need for solving a sequence of large-scale discretized, parametrized, partial diferential equations (PDEs) in the forward model. In this paper, we show how interpolatory parametric model reduction can significantly reduce the cost of the inversion process in DOT by drastically reducing the computational cost of solving the forward problems. The key observation is that function evaluations for the underlying optimization problem may be viewed as transfer function evaluations along the imaginary axis; a similar observation holds for Jacobian evaluations as well. This motivates the use of system-theoretic model order reduction methods. We discuss the construction and use of interpolatory parametric reduced models as surrogates for the full forward model. Within the DOT setting, these surrogate models can approximate both the cost functional and the associated Jacobian with very little loss of accuracy while significantly reducing the cost of the overall inversion process. Four numerical examples illustrate the effciency of the proposed approach. Although we focus on DOT in this paper, we believe that our approach is applicable much more generally. Published version

We introduce a numerical framework, the Voronoi Implicit Interface Method for tracking multiple interacting and evolving regions (phases) whose motion is determined by complex physics (fluids, mechanics, elasticity, etc.), intricate jump conditions, internal constraints, and boundary conditions. The method works in two and three dimensions, handles tens of thousands of interfaces and separate phases, and easily and automatically handles multiple junctions, triple points, and quadruple points in two dimensions, as well as triple lines, etc., in higher dimensions. Topological changes occur naturally, with no surgery required. The method is first-order accurate at junction points/lines, and of arbitrarily high-order accuracy away from such degeneracies. The method uses a single function to describe all phases simultaneously, represented on a fixed Eulerian mesh. We test the method’s accuracy through convergence tests, and demonstrate its applications to geometric flows, accurate prediction of von Neumann’s law for multiphase curvature flow, and robustness under complex fluid flow with surface tension and large shearing forces.

The level set method is a popular technique for tracking moving interfaces in several disciplines including computer vision and fluid dynamics. However, despite its high flexibility, the original level set method is limited by two important numerical issues. Firstly, the level set method does not implicitly preserve the level set function as a distance function, which is necessary to estimate accurately geometric features s.a. the curvature or the contour normal. Secondly, the level set algorithm is slow because the time step is limited by the standard CFL condition, which is also essential to the numerical stability of the iterative scheme. Recent advances with graph cut methods and continuous convex relaxation methods provide powerful alternatives to the level set method for image processing problems because they are fast, accurate and guaranteed to find the global minimizer independently to the initialization. These recent techniques use binary functions to represent the contour rather than distance functions, which are usually considered for the level set method. However, the binary function cannot provide the distance information, which can be essential for some applications s.a. the surface reconstruction problem from scattered points and the cortex segmentation problem in medical imaging. In this paper, we propose a fast algorithm to preserve distance functions in level set methods. Our algorithm is inspired by recent efficient $\ell^1$ optimization techniques, which will provide an efficient and easy to implement algorithm. It is interesting to note that our algorithm is not limited by the CFL condition and it naturally preserves the level set function as a distance function during the evolution, which avoids the classical re-distancing problem in level set methods. We apply the proposed algorithm to carry out image segmentation, where our methods proves to be 5 to 6 times faster than standard distance preserving level set techniques. We also present two applications where preserving a distance function is essential. Nonetheless, our method stays generic and can be applied to any level set methods that require the distance information.

[1] The breaking of free surface waves is investigated numerically via a Navier-Stokes model for the two-fluids flow of air and water. Third order Stokes waves in a periodic domain are simulated. The fundamental wavelength is 27 cm, whereas the initial steepness varies from low values, leading to regular wave trains, up to artificially steep wave trains yielding plunging breaking events. Attention is focused on the early stage of the breaking, when most of the energy is dissipated. The energy content in air, the fraction associated to surface tension effects, the viscous dissipation in water, and the work done against the pressure field are analyzed in order to distinguish the different contributions to the dissipation. Vorticity fields and dissipation contours are also presented. In the spilling case, the extra energy content with respect to the steepest nonbreaking case focusses into the breaking region and is gradually dissipated. Once the extra energy has been dissipated, the resulting wave matches the steepest nonbreaking solution. In the plunging case, an important role is played by the air entrainment. A fraction between 10 to 35% of the energy dissipated by the breaking is spent in entraining the air cavity, and most of it is dissipated by viscous effects when the cavity collapses. The phenomenon is clearly highlighted by sequences of vorticity and dissipation contours. The circulation and the area of the cavity generated by the plunging of the jet are provided, and parametric dependencies are proposed.

Industrial glass blowing is an essential stage of manufacturing hollow glass containers, e.g. bottles, jars. A glass preform is brought into a mould and inflated with compressed air until it reaches the mould shape. A simulation model for blowing glass containers based on finite element methods, which adopts a level set method to track the glass-air interfaces, has previously been developed [1–3]. A considerable challenge in glass blowing is the inverse problem: to determine an optimal preform from the desired container shape. In previous work of the authors [4, 5] a numerical method was introduced for optimising the shape of the preform. The optimisation method described the shape of the preform by parametric curves, e.g. Bezier-curves or splines, and employed a modified Levenberg-Marquardt algorithm to find the optimal positions of the control points of the curves. A combined finite difference and Broyden method was used to compute the Jacobian of the residual with respect to changes in the positions of the control points. The objective of this paper is to perform an error analysis of the optimisation method previously introduced and to improve its accuracy and performance. The improved optimisation method is applied to modelled containers of industrial relevance, which shows its usefulness for practical applications.

This work is currently under development within the framework of an American-European project (Digimat Project). The paper details the development of some numerical tools dedicated to the digital representation of metallic materials structures, to the finite element modelling of the polycrystalline microstructure deformation under large strains and to the subsequent recrystallization. The level set method used for the description of the microstructure interfaces is shown to represent a common base to all these developments. © 2007 American Institute of Physics.

In this paper, we present a 3D geometric flow designed to segment the main core of fiber tracts in diffusion tensor magnetic resonance images. The fundamental assumption of our fiber segmentation technique is that adjacent voxels in a tract have similar properties of diffusion. The fiber segmentation is carried out with a front propagation algorithm constructed to fill the whole fiber tract. The front is a 3D surface that evolves with a propagation speed proportional to a measure indicating the similarity of diffusion between the tensors lying on the surface and their neighbors in the direction of propagation. We use a level set implementation to assure a stable and accurate evolution of the surface and to handle changes of topology of the surface during the evolution process. The fiber tract segmentation method does not need a regularized tensor field since the surface is automatically smoothed as it propagates. The smoothing is done by an intrinsic surface force, based on the minimal principal curvature. This segmentation can be used for obtaining quantitative measures of the diffusion in the fiber tracts and it can also be used for white matter registration and for surgical planning.

This work presents image processing methods for the quantitative analysis of pericardial fat by using cardiac CT imaging. Although pericardial fat has recently been shown to correlate with cardiovascular disease, cardiovascular risk factors and metabolic syndrome, many concerns still remain about the methods for measuring pericardial fat, its regional distribution on the myocardium and the accuracy and reproducibility of the evaluations. This paper addresses these issues, proposing a method for the analysis of single-frame 3D images obtained by the standard acquisition protocol used in coronary calcium scoring. Experimental results show that the method produces evaluations in good agreement with those obtained by manual segmentation. Thus, the introduced method may be used for the systematic evaluation of pericardial fat, that may be useful for risk stratification of patients undergoing cardiac CT.

A multiple marker level-set (MLS) method is introduced for Direct Numerical Simulation of bubbles and droplets, which is integrated in a finite-volume framework on collocated unstructured grids of arbitrary element type. The MLS method is based on the idea of describing separate interfaces in the disperse phase with different levelset functions to prevent numerical and potentially unphysical coalescence of bubbles or droplets without excessive refinement. Thus, bubbles or droplets are able to approach each other closely, within the size of one grid cell, and can even collide, while artificial merging of the interfaces is prevented. The accuracy of the computational method implemented is examined for the problem of a single bubble rising in quiescent liquid. For the validation of the multiple marker level-set method we study the gravity-driven impact of a droplet on a liquid-liquid interface and finally it is applied on simulation of gravity-driven bubbly flow in a vertical pipe.

The main purpose of this thesis is to propose a method for structural optimization which combines the accuracy of featuring an exact description of shapes (i.e. with a mesh) at each iteration of the process with the versatility of the level set method for tracking their evolution. Independently, we also study two problems related to modeling in structural optimization. In the first, bibliographical part, we present several classical notions, together with some recent developments about the three main issues of this thesis - namely level set methods (Chapter 1), shape optimization (Chapter 2), and meshing (Chapter 3). The second part of this manuscript deals with two issues in shape optimization, that of the optimal repartition of several materials within a fixed structure (Chapter 4), and that of the robust optimization of functions depending on the domain when perturbations are expected over the considered mechanical model. In the third part, we study the design of numerical schemes for performing the level set method on simplicial (and possibly adapted) computational meshes. The computation of the signed distance function to a domain is investigated in Chapter 6, and the resolution of the level set advection equation is presented in Chapter 7. The fourth part (Chapter 8) is devoted to the meshing techniques introduced in this thesis. Eventually, the last part (Chapter 9) describes the proposed strategy for mesh evolution in the context of shape optimization, relying on the numerical ingredients introduced in Chapters 7, 8, 9.; L'objectif principal de cette thèse est de concevoir une méthode d'optimisation de structures qui jouit d'une description exacte (i.e. au moyen d'un maillage) de la forme à chaque itération du processus, tout en bénéficiant des avantages de la méthode des lignes de niveaux lorsqu'il s'agit de suivre leur évolution. Indépendamment, on étudie également deux problèmes de modélisation en optimisation structurale. Dans une première partie bibliographique, on présente quelques notions classiques, ainsi qu'un état de l'art sommaire autour des trois thématiques principales de la thèse - méthode des lignes de niveaux (Chapitre 1), optimisation de formes (Chapitre 2) et maillage (Chapitre 3). La seconde partie de ce manuscrit traite de deux questions en optimisation de formes, celle de la répartition optimale de plusieurs matériaux au sein d'une structure donnée (Chapitre 4), et celle de l'optimisation robuste de fonctions dépendant du domaine lorsque des perturbations s'exercent sur le modèle (Chapitre 5). Dans une troisième partie, on étudie la conception de schémas numériques en lien avec la méthode des lignes de niveaux lorsque le maillage de calcul est simplicial (et potentiellement adapté). Le calcul de la distance signée à un domaine est étudié dans le chapitre 6, et la résolution de l'équation de transport d'une fonction 'level set' est détaillée dans le chapitre 7. La quatrième partie (Chapitre 8) traite des aspects de la thèse liés à la modification locale de maillages surfaciques et volumiques. Enfin, la dernière partie (Chapitre 9) détaille la stratégie conçue pour l'évolution de maillage en optimisation de formes, à partir des ingrédients des chapitres 6, 7 et 8.

A level-set model is presented for simulating mass transfer or heat transfer in two-phase flows. The Navier-Stokes equations and mass transfer (or heat transfer) equation are discretized using a finite volume method on a collocated unstructured mesh, whereas a multiple marker level-set approach is used for interface capturing in bubble swarms. This method avoids the numerical coalescence of the fluid particles, whereas the mass conservation issue inherent to standard level-set methods is circumvented. Furthermore, unstructured flux-limiter schemes are used to discretize the convective term of momentum transport equation, level-set equations, and chemical species concentration equation, to avoid numerical oscillations around discontinuities, and to minimize the numerical diffusion. A convection-diffusion-reaction equation is used as a mathematical model for the chemical species mass transfer at the continuous phase. Because the mathematical analogy between dilute mass transfer and heat transfer, the same numerical model is applicable to solve both phenomena. The capabilities of this model are proved for the diffusion of chemical species from a sphere, external mass transfer in the buoyancy-driven motion of single bubbles and bubble swarms. Results are extensively validated by comparison with analytical solutions and empirical correlations from the literature.

In this work we consider 3D point sets, which in a typical setting represent unorganized point clouds. Segmentation of these point sets requires first to single out structural components of the unknown surface discretely approximated by the point cloud. Structural components, in turn, are surface patches approximating unknown parts of elementary geometric structures, such as planes, ellipsoids, spheres and so on. The approach used is based on level set methods computing the moving front of the surface and tracing the interfaces between different parts of it. Level set methods are widely recognized to be one of the most efficient methods to segment both 2D images and 3D medical images. Level set methods for 3D segmentation have recently received an increasing interest. We contribute by proposing a novel approach for raw point sets. Based on the motion and distance functions of the level set we introduce four energy minimization models, which are used for segmentation, by considering an equal number of distance functions specified by geometric features. Finally we evaluate the proposed algorithm on point sets simulating unorganized point clouds.