Publications

2020

• Modelling the fluid-structure coupling caused by a far-field underwater explosion
  
  • Mavaleix-Marchessoux Damien
  , 2020. Submarines must withstand the effects of rapid dynamic loads induced by underwater explosions. Due to the very high cost of experimental campaigns, numerical simulations are very important. A remote underwater explosion is a complex event that has two distinct effects: it sends a shock wave, then creates an oscillating gas bubble that sets water in slower motion. The two phenomena have quite different characteristics and time scales. In this work, we consider remote enough underwater explosions so that (i) the presence of the submarine only marginally affects the explosion, and (ii) there is a temporal separation of the two phenomena, as experienced by the ship. Under these conditions, our overall goal is to design, implement (in the context of high performance computing) then validate a computational methodology for the fluid-structure interaction problem, taking into account both phenomena. With this aim, we first study the two perturbations without considering the submarine, to propose appropriate modelling and numerical methods. Then, we design a fast boundary element (BEM) procedure, based on the combination of the convolution quadrature method and an original empirical high frequency approximation. The procedure allows to efficiently simulate 3D rapid transient wave propagation problems set in an unbounded domain, and shows advantageous complexity: O(1) in regards to the time discretisation and O(N log N) for the spatial discretisation. Finally, we implement adequate finite element/boundary element (FEM/BEM) coupling strategies for the shock wave fluid-structure interaction phase (linear acoustics) and that of the gas bubble (incompressible flow). The overall procedure, validated on academic problems, provides very promising results when applied on realistic industrial cases.
• An analytical approach based on tailored Green's functions for flow noise prediction at low Mach number
  
  • Trafny Nicolas
  • Serre Gilles
  • Cotte Benjamin
  • Mercier Jean-François
  , 2020, pp.749-750. The presence of boundary surfaces in a turbulent flow can result in the enhancement of the radiated acoustic field especially for eddies close to any geometrical singularity. At low Mach number, it is well known that the contribution of the diffracted field is dominant. In the present study, we focus on Lighthill?s wave equation solved using a tailored Green?s function and a semi-empirical turbulence model in order to investigate the direct acoustic field produced by a turbulent boundary layer over a flat plate. We consider both the turbulent boundary layer noise and edge noise, and power law results deduced from classical dimensional analysis are recovered. The specular acoustic field, produced by eddies far from any edge increases in proportion to the fourth power of the Mach number and the diffracted field directly in proportion to the Mach number. Finally, we chose a NACA 0012 airfoil in order to validate the noise spectrum prediction for different configurations for which trailing edge noise is the dominant contribution. (10.48465/fa.2020.0912)
  DOI : 10.48465/fa.2020.0912
• Non-overlapping domain decomposition methods with non-local transmission operators for harmonic wave propagation problems
  
  • Parolin Émile
  , 2020. The pioneering work of B. Després then M. Gander, F. Magoulès and F. Nataf have shown that it is mandatory, at least in the context of wave equations, to use impedance type transmission conditions in the coupling of subdomains in order to obtain convergence of non-overlapping domain decomposition methods (DDM). In the standard approach considered in the literature, the impedance operator involved in the transmission conditions is local and leads to algebraic convergence of the DDM in the best cases. In later works, F. Collino, S. Ghanemi and P. Joly then F. Collino, P. Joly and M. Lecouvez have observed that using non local impedance operators such as integral operators with suitable singular kernels could lead to a geometric convergence of the DDM.This thesis extends these works (that mainly concerned the scalar Helmholtz equation) with the extension of the analysis to electromagnetic wave propagation. Besides, the numerical analysis of the method is performed for the first time, proving the stability of the convergence rate with respect to the discretization parameter, hence the robustness of the approach. Several integral operators are then proposed as transmission operators for Maxwell equations in the spirit of those constructed for the acoustic setting. An alternative to integral operators, based on the resolution of elliptic auxiliary problems, is also advocated and analyzed. Extensive numerical results are conducted, illustrating the high potential of the new approach. Based on a recent work by X. Claeys, the last part of this work consists in exploiting the multi-trace formalism to extend the convergence analysis to the case of partitions with junction points, which is a difficult problem that attracted a lot of attention recently. The new approach relies on a new operator that communicates information between sub-domains, which replaces the classical point-to-point exchange operator. A proof of geometrical convergence of the associated iterative algorithm, again uniform with respect to the discretization parameter, is available and we show that one recovers the standard algorithm in the absence of junction points.
• A fast boundary element based solver for localized inelastic deformations
  
  • Ciardo Federico
  • Lecampion Brice
  • Fayard François
  • Chaillat Stéphanie
  International Journal for Numerical Methods in Engineering, Wiley, 2020, 121 (24), pp.5696 - 5718. We present a numerical method for the solution of nonlinear geomechanical problems involving localized deformation along shear bands and fractures. We leverage the boundary element method to solve for the quasi-static elastic deformation of the medium while rigid-plastic constitutive relations govern the behavior of displacement discontinuity (DD) segments capturing localized deformations. A fully implicit scheme is developed using a hierarchical approximation of the boundary element matrix. Combined with an adequate block preconditioner, this allows to tackle large problems via the use of an iterative solver for the solution of the tangent system. Several two-dimensional examples of the initiation and growth of shear-bands and tensile fractures illustrate the capabilities and accuracy of this technique. The method does not exhibit any mesh dependency associated with localization provided that (i) the softening length-scale is resolved and (ii) the plane of localized deformations is discretized a priori using DD segments. (10.1002/nme.6520)
  DOI : 10.1002/nme.6520
• Computation of the exact discrete transparent boundary condition for 1D linear equations
  
  • Fliss Sonia
  • Imperiale Sébastien
  • Tonnoir Antoine
  , 2020. In this work, we are interested in the construction of the exact transparent boundary conditions for a semi-discretized and fully discretized 1D linear PDE. The proposed method is quite general and is based on the computation of a family of canonical functions. Several examples and numerical results to illustrate the method are presented.
• A Discrete Domain Decomposition Method for Acoustics with Uniform Exponential Rate of Convergence Using Non-local Impedance Operators
  
  • Claeys Xavier
  • Collino Francis
  • Joly Patrick
  • Parolin Emile
  , 2020, pp.310-317. The relaxed Jacobi algorithm written at the continuous level was proven to converge exponentially. However, it was only a conjecture, hinted at by numerical experiments in [5, Section 8], that the discretized algorithm using finite elements has a rate of convergence uniformly bounded with respect to the discretization parameter (10.1007/978-3-030-56750-7_35)
  DOI : 10.1007/978-3-030-56750-7_35
• Analyse spectrale et simulation numérique de cavités contenant un matériau négatif
  
  • Bernard Paolantoni Sandrine
  , 2020. Cette thèse réalise une étude théorique et numérique du spectre de cavités partiellement composées de matériau négatif, c'est-à-dire de matériau pour lequel la perméabilité magnétique et/ou la permittivité électrique (ou au moins leur partie réelle) deviennent négatives dans certaines plages de fréquences. Cette étude s'inscrit dans la continuité des travaux engagés dans notre laboratoire qui se concentrent sur la propagation des ondes électromagnétiques en présence de matériau négatif, à fréquence fixée. L'objectif de cette thèse est de prendre en compte la dispersion fréquentielle, autrement dit la dépendance en fréquence de la perméabilité et de la permittivité, en considérant la fréquence comme paramètre spectral. Nous mettons en évidence le spectre essentiel résultant de la présence de matériau négatif ainsi que les phénomènes de résonance qui en découlent, pour différents modèles décrivant ce matériau.L'étude théorique se concentre sur le cas de cavités bidimensionnelles polygonales pour les modèles de Drude et de Lorentz (avec et sans dissipation). L'étude théorique du modèle le plus simple (Drude non dissipatif) est étendue au cas d'une interface courbe (mais régulière).Ce modèle fait également l'objet d'une étude numérique, visant à explorer l'effet d'une discrétisation éléments finis du problème théorique, et ainsi mettre en avant les difficultés à observer numériquement certains des phénomènes de résonance.
• Asymptotic modelling of Skin-effects in coaxial cables
  
  • Beck Geoffrey
  • Imperiale Sébastien
  • Joly Patrick
  SN Partial Differential Equations and Applications, Springer, 2020. In this work we tackle the modeling of non-perfectly conducting thin coaxial cables. From the non-dimensionnalised 3D Maxwells equations, we derive, by asymptotic analysis with respect to the (small) transverse dimension of the cable, a simplified effective 1D model and an effective reconstruction procedure of the electric and magnetic fields. The derived effective model involves a fractional time derivatives that accounts for the so-called skin effects in highly conducting regions.
• Experimental and theoretical observations on DDT in smooth narrow channels
  
  • Melguizo-Gavilanes J.
  • Ballossier Yves
  • Maltez Faria Luiz
  Proceedings of the Combustion Institute, Elsevier, 2020. A combined experimental and theoretical study of deflagration-to-detonation transition (DDT) in smooth narrow channels is presented. Some of the distinguishing features characterizing the late stages of DDT are shown to be qualitatively captured by a simple one-dimensional scalar equation. Inspection of the structure and stability of the traveling wave solutions found in the model, and comparison with experimental observations, suggest a possible mechanism responsible for front acceleration and transition to detonation. (10.1016/j.proci.2020.07.142)
  DOI : 10.1016/j.proci.2020.07.142
• Time harmonic wave propagation in one dimensional weakly randomly perturbed periodic media
  
  • Fliss Sonia
  • Giovangigli Laure
  SN Partial Differential Equations and Applications, Springer, 2020, 1 (40). In this work we consider the solution of the time harmonic wave equation in a one dimensional periodic medium with weak random perturbations. More precisely, we study two types of weak perturbations: (1) the case of stationary, ergodic and oscillating coefficients, the typical size of the oscillations being small compared to the wavelength and (2) the case of rare random perturbations of the medium, where each period has a small probability to have its coefficients modified, independently of the other periods. Our goal is to derive an asymptotic approximation of the solution with respect to the small parameter. This can be used in order to construct absorbing boundary conditions for such media.
• Transparent boundary conditions for wave propagation in fractal trees: convolution quadrature approach
  
  • Joly Patrick
  • Kachanovska Maryna
  Numerische Mathematik, Springer Verlag, 2020, 146(2), pp.281-334. In this work we propose high-order transparent boundary conditions for the weighted wave equation on a fractal tree, with an application to the modeling of sound propagation in a human lung. This article follows the recent work [29], dedicated to the mathematical analysis of the corresponding problem and the construction of low-order absorbing boundary conditions. The method proposed in this article consists in constructing the exact (trans-parent) boundary conditions for the semi-discretized problem, in the spirit of the convolution quadrature method developed by Ch. Lubich. We analyze the stability and convergence of the method, and propose an efficient algorithm for its implementation. The exposition is concluded with numerical experiments.
• A non-overlapping domain decomposition method with high-order transmission conditions and cross-point treatment for Helmholtz problems
  
  • Modave Axel
  • Royer Anthony
  • Antoine Xavier
  • Geuzaine Christophe
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2020, 368, pp.1131622020. A non-overlapping domain decomposition method (DDM) is proposed for the parallel finite-element solution of large-scale time-harmonic wave problems. It is well-known that the convergence rate of this kind of method strongly depends on the transmission condition enforced on the interfaces between the subdomains. Local conditions based on high-order absorbing boundary conditions (HABCs) have proved to be well-suited, as a good compromise between basic impedance conditions, which lead to suboptimal convergence, and conditions based on the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain — which are too expensive to compute. However, a direct application of this approach for configurations with interior cross-points (where more than two subdomains meet) and boundary cross-points (points that belong to both the exterior boundary and at least two subdomains) is suboptimal and, in some cases, can lead to incorrect results. In this work, we extend a non-overlapping DDM with HABC-based transmission conditions approach to efficiently deal with cross-points for lattice-type partitioning. We address the question of the cross-point treatment when the HABC operator is used in the transmission condition, or when it is used in the exterior boundary condition, or both. The proposed cross-point treatment relies on corner conditions developed for Padé-type HABCs. Two-dimensional numerical results with a nodal finite-element discretization are proposed to validate the approach, including convergence studies with respect to the frequency, the mesh size and the number of subdomains. These results demonstrate the efficiency of the cross-point treatment for settings with regular partitions and homogeneous media. Numerical experiments with distorted partitions and smoothly varying heterogeneous media show the robustness of this treatment. (10.1016/j.cma.2020.113162)
  DOI : 10.1016/j.cma.2020.113162
• On the efficiency of nested GMRES preconditioners for 3D acoustic and elastodynamic H-matrix accelerated Boundary Element Methods
  
  • Kpadonou Félix D.
  • Chaillat Stéphanie
  • Ciarlet Patrick
  Computers & Mathematics with Applications, Elsevier, 2020, 80 (3). This article is concerned with the derivation of fast Boundary Element Methods for 3D acoustic and elastodynamic problems. In particular, we are interested in the acceleration of Hierarchical matrix (-matrix) based iterative solvers. While H-matrix representations allow to reduce the storage requirements and the cost of a matrix–vector product, the number of iterations for an iterative solver, as the frequency or the problem size increases, remains an issue. We consider an inner–outer preconditioning strategy, i.e., the preconditioner is applied through an iterative solver at the inner level. The preconditioner is defined as a H-matrix representation of the system matrix with a given accuracy. We investigate the influence of various parameters of the preconditioner, i.e., the H-matrix accuracy, the GMRES threshold and the maximum number of iterations of the inner solver. Different numerical results are presented to compare the efficiency of the preconditioner with respect to the unpreconditioned reference system. Finally, we propose a way to define the optimal setting for this preconditioner. (10.1016/j.camwa.2020.03.021)
  DOI : 10.1016/j.camwa.2020.03.021
• Planewave Density Interpolation Methods for the EFIE on Simple and Composite Surfaces
  
  • Pérez-Arancibia Carlos
  • Turc Catalin
  • Faria Luiz
  • Sideris Constantine
  IEEE Transactions on Antennas and Propagation, Institute of Electrical and Electronics Engineers, 2020. This paper presents an extension of the recently introduced planewave density interpolation (PWDI) method to the electric field integral equation (EFIE) formulation of problems of scattering and radiation by perfect electric conducting (PEC) objects. Relying on Kirchhoff integral formula and local interpolation of surface current densities that regularize the kernel singularities, the PWDI method enables off-and on-surface EFIE operators to be re-expressed in terms of integrands that are globally bounded (or even more regular) over the whole domain of integration, regardless of the magnitude of the distance between target and source points. Surface integrals resulting from the application of the method-of-moments (MoM) using Rao-Wilton-Glisson (RWG) basis functions, can then be directly and easily evaluated by means of elementary quadrature rules irrespective of the singularity location. The proposed technique can be applied to simple and composite surfaces comprising two or more simply-connected overlapping components. The use of composite surfaces can significantly simplify the geometric treatment of complex structures, as the PWDI method enables the use of separate non-conformal meshes for the discretization of each of the surface components that make up the composite surface. A variety of examples, including multi-scale and intricate structures, demonstrate the effectiveness of the proposed methodology.
• Exponentially convergent non overlapping domain decomposition methods for the Helmholtz equation
  
  • Collino Francis
  • Joly Patrick
  • Lecouvez Matthieu
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2020, 54 (3), pp.775-810. In this paper, we develop in a general framework a non overlapping Domain Decomposition Method that is proven to be well-posed and converges exponentially fast, provided that specific transmission operators are used. These operators are necessarily non local and we provide a class of such operators in the form of integral operators. To reduce the numerical cost of these integral operators, we show that a truncation process can be applied that preserves all the properties leading to an exponentially fast convergent method. A modal analysis is performed on a separable geometry to illustrate the theoretical properties of the method and we exhibit an optimization process to further reduce the convergence rate of the algorithm. (10.1051/m2an/2019050)
  DOI : 10.1051/m2an/2019050
• The Linear Sampling Method for Kirchhoff-Love Infinite Plates
  
  • Bourgeois Laurent
  • Recoquillay Arnaud
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2020, 14 (2), pp.363-384. This paper addresses the problem of identifying impenetrable obstacles in a Kirchhoff-Love infinite plate from multistatic near-field data. The Linear Sampling Method is introduced in this context. We firstly prove a uniqueness result for such an inverse problem. We secondly provide the classical theoretical foundation of the Linear Sampling Method. We lastly show the feasibility of the method with the help of numerical experiments. (10.3934/ipi.2020016)
  DOI : 10.3934/ipi.2020016
• Adaptive solution of the neutron diffusion equation with heterogeneous coefficients using the mixed finite element method on structured meshes
  
  • Do M.-H.
  • Madiot F.
  • Ciarlet Patrick
  , 2020.
• On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates
  
  • Bourgeois Laurent
  • Chesnel Lucas
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2020. We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter ε > 0. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in ε. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due to the particular structure of the regularized problems, classical techniques à la Grisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence in ε in all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework. (10.1051/m2an/2019073)
  DOI : 10.1051/m2an/2019073
• Perfect Brewster transmission through ultrathin perforated films
  
  • Pham Kim
  • Maurel Agnès
  • Mercier Jean-François
  • Félix Simon
  • Cordero Maria Luisa
  • Horvath Camila
  Wave Motion, Elsevier, 2020, 93, pp.102485. We address the perfect transmission of a plane acoustic wave at oblique incidence on a perforated, sound penetrable or rigid, film in two-dimensions. It is shown that the Brewster incidence θ * realizing so-called extraordinary transmission due to matched impedances varies significantly when the thickness e of the film decreases. For thick films, i.e. ke ≫ 1 with k the incident wavenumber, the classical effective medium model provides an accurate prediction of the Brewster angle independent of e (this Brewster angle is denoted θ B). However, for thinner films with ke < 1, θ * becomes dependent of e and it deviates from θ B. To properly describe this shift, an interface model is used (Marigo et al., 2017) which accurately reproduces the spectra of ultrathin to relatively thick perforated films. Depending on the contrasts in the material properties of the film and of the surrounding matrix, decreasing the film thickness can produce an increase or a decrease of θ * ; it can also produce the disappearance of a perfect transmission or to the contrary its appearance. (10.1016/j.wavemoti.2019.102485)
  DOI : 10.1016/j.wavemoti.2019.102485
• Degenerate elliptic equations for resonant wave problems
  
  • Nicolopoulos Anouk
  • Campos Pinto Martin
  • Després Bruno
  • Ciarlet Patrick
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2020, 85 (1), pp.132-159. The modeling of resonant waves in 2D plasma leads to the coupling of two degenerate elliptic equations with a smooth coeffcient alpha and compact terms. The coeffcient alpha changes sign. The region where alpha is positive is propagative, and the region where alpha is negative is non propagative and elliptic. The two models are coupled through the line Sigma, corresponding to alpha equal to zero. Generically, it is an ill-posed problem, and additional information must be introduced to get a satisfactory treatment at Sigma. In this work we define the solution by relying on the limit absorption principle (alpha is replaced by alpha + i0^+) in an adapted functional setting. This setting lies on the decomposition of the solution in a regular part and a singular part, which originates at Sigma, and on quasi-solutions. It leads to a new well-posed mixed variational formulation with coupling. As we design explicit quasi-solutions, numerical experiments can be carried out, which illustrate the good properties of this new tool for numerical computation. (10.1093/imamat/hxaa001)
  DOI : 10.1093/imamat/hxaa001
• A fast boundary element method using the Z-transform and high-frequency approximations for large-scale 3D transient wave problems
  
  • Mavaleix-Marchessoux Damien
  • Bonnet Marc
  • Chaillat Stéphanie
  • Leblé Bruno
  International Journal for Numerical Methods in Engineering, Wiley, 2020, 121, pp.4734-4767. 3D rapid transient acoustic problems are difficult to solve numerically when dealing with large geometries, because numerical methods based on geometry discretisation (mesh), such as the boundary element method (BEM) or the finite element method (FEM), often require to solve a linear system (from the spacial discretisation) for each time step. We propose a numerical method to efficiently deal with 3D rapid transient acoustic problems set in large exterior domains. Using the Z-transform and the convolution quadrature method (CQM), we first present a straightforward way to reframe the problem to the solving of a large amount (the number of time steps, M) of frequency-domain BEMs. Then, taking advantage of a well-designed high-frequency approximation (HFA), we drastically reduce the number of frequency-domain BEMs to be solved, with little loss of accuracy. The complexity of the resulting numerical procedure turns out to be O(1) in regards to the time discretisation and O(N log N) for the spacial discretisation, the latter being prescribed by the complexity of the used fast BEM solver. Examples of applications are proposed to illustrate the efficiency of the procedure in the case of fluid-structure interaction: the radiation of an acoustic wave into a fluid by a deformable structure with prescribed velocity, and the scattering of an abrupt wave by simple and realistic geometries. (10.1002/nme.6488)
  DOI : 10.1002/nme.6488
• Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering
  
  • Modave Axel
  • Geuzaine Christophe
  • Antoine Xavier
  Journal of Computational Physics, Elsevier, 2020, 401, pp.109029. This paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal and polyhedral computational domains for the finite element solution of high-frequency acoustic scattering problems. While high-order absorbing boundary conditions (HABCs) are accurate for smooth fictitious boundaries, the precision of the solution drops in the presence of corners if no specific treatment is applied. We present and analyze two strategies to preserve the accuracy of Padé-type HABCs at corners: first by using compatibility relations (derived for right angle corners) and second by regularizing the boundary at the corner. Exhaustive numerical results for two- and three-dimensional problems are reported in the paper. They show that using the compatibility relations is optimal for domains with right angles. For the other cases, the error still remains acceptable, but depends on the choice of the corner treatment according to the angle. (10.1016/j.jcp.2019.109029)
  DOI : 10.1016/j.jcp.2019.109029
• On well-posedness of scattering problems in a Kirchhoff-Love infinite plate
  
  • Bourgeois Laurent
  • Hazard Christophe
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2020, 80 (3), pp.1546-1566. We address scattering problems for impenetrable obstacles in an infinite elastic Kirchhoff-Love two-dimensional plate. The analysis is restricted to the purely bending case and the time-harmonic regime. Considering four types of boundary conditions on the obstacle, well-posedness for those problems is proved with the help of a variational approach: (i) for any wave number k when the plate is clamped, simply supported or roller supported; (ii) for any k except a discrete set when the plate is free (this set is finite for convex obstacles).
• Modified forward and inverse Born series for the Calderon and diffuse-wave problems
  
  • Abhishek Anuj
  • Bonnet Marc
  • Moskow Shari
  Inverse Problems, IOP Publishing, 2020, 36, pp.114001. We propose a new direct reconstruction method based on series inversion for Electrical Impedance Tomography (EIT) and the inverse scattering problem for diffuse waves. The standard Born series for the forward problem has the limitation that the series requires that the contrast lies within a certain radius for convergence. Here, we instead propose a modified Born series which converges for the forward problem unconditionally. We then invert this modified Born series and compare reconstructions with the usual inverse Born series. We also show that the modified inverse Born series has a larger radius of convergence. (10.1088/1361-6420/abae11)
  DOI : 10.1088/1361-6420/abae11
• Shape optimization of Stokesian peristaltic pumps using boundary integral methods
  
  • Bonnet Marc
  • Liu Ruowen
  • Veerapaneni Shravan
  Journal of Computational and Applied Mathematics, Elsevier, 2020, 46, pp.18. This article presents a new boundary integral approach for finding optimal shapes of peristaltic pumps that transport a viscous fluid. Formulas for computing the shape derivatives of the standard cost functionals and constraints are derived. They involve evaluating physical variables (traction, pressure, etc.) on the boundary only.By employing these formulas in conjunction with a boundary integral approach for solving forward and adjoint problems, we completely avoid the issue of volume remeshing when updating the pump shape as the optimization proceeds. This leads to significant cost savings and we demonstrate the performance on several numerical examples (10.1007/s10444-020-09761-7)
  DOI : 10.1007/s10444-020-09761-7