Performance of interFoam on the simulation of progressive waves

The performance of interFoam (a widely-used solver within the popular open source CFD package OpenFOAM) in simulating the propagation of a nonlinear (stream function solution) regular wave is investigated in this work, with the aim of systematically documenting its accuracy. It is demonstrated that over time there is a tendency for surface elevations to increase, wiggles to appear in the free surface, and crest velocities to become (severely) overestimated. It is shown that increasing the temporal and spatial resolution can mitigate these undesirable effects, but that a relatively small Courant number is required. It is further demonstrated that the choice of discretization schemes and solver settings (often treated as a "black box" by users) can have a major impact on the results. This impact is documented, and it is shown that obtaining a "diffusive balance" is crucial to accurately propagate a surface wave over long distances without requiring exceedingly high temporal and spatial resolutions. Finally, the new code isoAdvector is compared to interFoam, which is demonstrated to produce comparably accurate results, while maintaining a sharper surface. It is hoped that the systematic documentation of the performance of the interFoam solver will enable its more accurate and optimal use, as well as increase awareness of potential shortcomings, by CFD researchers interested in the general CFD simulation of free surface waves.Comment: 18 pages and 23 figure

A new numerical strategy with space-time adaptivity and error control for multi-scale streamer discharge simulations

This paper presents a new resolution strategy for multi-scale streamer discharge simulations based on a second order time adaptive integration and space adaptive multiresolution. A classical fluid model is used to describe plasma discharges, considering drift-diffusion equations and the computation of electric field. The proposed numerical method provides a time-space accuracy control of the solution, and thus, an effective accurate resolution independent of the fastest physical time scale. An important improvement of the computational efficiency is achieved whenever the required time steps go beyond standard stability constraints associated with mesh size or source time scales for the resolution of the drift-diffusion equations, whereas the stability constraint related to the dielectric relaxation time scale is respected but with a second order precision. Numerical illustrations show that the strategy can be efficiently applied to simulate the propagation of highly nonlinear ionizing waves as streamer discharges, as well as highly multi-scale nanosecond repetitively pulsed discharges, describing consistently a broad spectrum of space and time scales as well as different physical scenarios for consecutive discharge/post-discharge phases, out of reach of standard non-adaptive methods.Comment: Support of Ecole Centrale Paris is gratefully acknowledged for several month stay of Z. Bonaventura at Laboratory EM2C as visiting Professor. Authors express special thanks to Christian Tenaud (LIMSI-CNRS) for providing the basis of the multiresolution kernel of MR CHORUS, code developed for compressible Navier-Stokes equations (D\'eclaration d'Invention DI 03760-01). Accepted for publication; Journal of Computational Physics (2011) 1-2

Mesh-based video coding for low bit-rate communications

In this paper, a new method for low bit-rate content-adaptive mesh-based video coding is proposed. Intra-frame coding of this method employs feature map extraction for node distribution at specific threshold levels to achieve higher density placement of initial nodes for regions that contain high frequency features and conversely sparse placement of initial nodes for smooth regions. Insignificant nodes are largely removed using a subsequent node elimination scheme. The Hilbert scan is then applied before quantization and entropy coding to reduce amount of transmitted information. For moving images, both node position and color parameters of only a subset of nodes may change from frame to frame. It is sufficient to transmit only these changed parameters. The proposed method is well-suited for video coding at very low bit rates, as processing results demonstrate that it provides good subjective and objective image quality at a lower number of required bits

Iso-geometric Integral Equation Solvers and their Compression via Manifold Harmonics

The state of art of electromagnetic integral equations has seen significant growth over the past few decades, overcoming some of the fundamental bottlenecks: computational complexity, low frequency and dense discretization breakdown, preconditioning, and so on. Likewise, the community has seen extensive investment in development of methods for higher order analysis, in both geometry and physics. Unfortunately, these standard geometric descriptors are $C^0$ at the boundary between patches with a few exceptions; as a result, one needs to define additional mathematical infrastructure to define physical basis sets for vector problems. In stark contrast, the geometric representation used for design is higher-order differentiable over the entire surface. Geometric descriptions that have $C^{2}$-continuity almost everywhere on the surfaces are common in computer graphics. Using these description for analysis opens the door to several possibilities, and is the area we explore in this paper. Our focus is on Loop subdivision based isogeometric methods. In this paper, our goals are two fold: (i) development of computational infrastructure necessary to effect efficient methods for isogeometric analysis of electrically large simply connected objects, and (ii) to introduce the notion of manifold harmonics transforms and its utility in computational electromagnetics. Several results highlighting the efficacy of these two methods are presented

Monotonicity-preserving finite element methods for hyperbolic problems

This thesis covers the development of monotonicity preserving finite element methods for hyperbolic problems. In particular, scalar convection-diffusion and Euler equations are used as model problems for the discussion in this dissertation. A novel artificial diffusion stabilization method has been proposed for scalar problems. This technique is proved to yield monotonic solutions, to be \ac{led}, Lipschitz continuous, and linearity preserving. These properties are satisfied in multiple dimensions and for general meshes. However, these results are limited to first order Lagrangian finite elements. A modification of this stabilization operator that is twice differentiable has been also proposed. With this regularized operator, nonlinear convergence is notably improved, while the stability properties remain unaltered (at least, in a weak sense). An extension of this stabilization method to high-order discretizations has also been proposed. In particular, arbitrary order space-time isogeometric analysis is used for this purpose. It has been proved that this scheme yields solutions that satisfy a global space-time discrete maximum principle unconditionally. A partitioned approach has also been proposed. This strategy reduces the computational cost of the scheme, while it preserves all stability properties. A regularization of this stabilization operator has also been developed. As for the first order finite element method, it improves the nonlinear convergence without harming the stability properties. An extension to Euler equations has also been pursued. In this case, instead of monotonicity-preserving, the developed scheme is local bounds preserving. Following the previous works, a regularized differentiable version has also been proposed. In addition, a continuation method using the parameters introduced for the regularization has been used. In this case, not only the nonlinear convergence is improved, but also the robustness of the method. However, the improvement in nonlinear convergence is limited to moderate tolerances and it is not as notable as for the scalar problem. Finally, the stabilized schemes proposed had been adapted to adaptive mesh refinement discretizations. In particular, nonconforming hierarchical octree-based meshes have been used. Using these settings, the efficiency of solving a monotonicity-preserving high-order stiff nonlinear problem has been assessed. Given a specific accuracy, the computational time required for solving the high-order problem is compared to the one required for solving a low-order problem (easy to converge) in a much finer adapted mesh. In addition, an error estimator based on the stabilization terms has been proposed and tested. The performance of all proposed schemes has been assessed using several numerical tests and solving various benchmark problems. The obtained results have been commented and included in the dissertation.La present tesi tracta sobre mètodes d'elements finits que preserven la monotonia per a problemes hiperbòlics. Concretament, els problemes que s'han utilitzat com a model en el desenvolupament d'aquesta tesi són l'equació escalar de convecció-difusió-reacció i les equacions d'Euler. Per a problemes escalars s'ha proposat un nou mètode d'estabilització mitjançant difusió artificial. S'ha provat que amb aquesta tècnica les solucions obtingudes són monòtones, l'esquema "disminueix els extrems locals", i preserva la linearitat. Aquestes propietats s'han pogut demostrar per múltiples dimensions i per malles generals. Per contra, aquests resultats només són vàlids per elements finits Lagrangians de primer ordre. També s'ha proposat una modificació de l'operador d'estabilització per tal de que aquest sigui diferenciable. Aquesta regularització ha permès millorar la convergència no-lineal notablement, mentre que les propietats d'estabilització no s'han vist alterades. L'anterior mètode d'estabilització s'ha adaptat a discretitzacions d'alt ordre. Concretament, s'ha utilitzat anàlisi isogeomètrica en espai i temps per a aquesta tasca. S'ha provat que les solucions obtingudes mitjançant aquest mètode satisfan el principi del màxim discret de forma global. També s'ha proposat un esquema particionat. Aquesta alternativa redueix el cost computacional, mentre preserva totes les propietats d'estabilitat. En aquest cas, també s'ha realitzat una regularització de l'operador d'estabilització per tal de que sigui diferenciable. Tal i com s'ha observat en els mètodes de primer ordre, aquesta regularització permet millorar la convergència no-lineal sense perdre les propietats d'estabilització. Posteriorment, s'ha estudiat l'adaptació dels mètodes anteriors a les equacions d'Euler. En aquest cas, en comptes de preservar la monotonia, l'esquema preserva "cotes locals". Seguint els desenvolupaments anteriors, s'ha proposat una versió diferenciable de l'estabilització. En aquest cas, també s'ha desenvolupat un mètode de continuació utilitzant els paràmetres introduïts per a la regularització. En aquest cas, no només ha millorat la convergència no-lineal sinó que l'esquema també esdevé més robust. Per contra, la millora en la convergència no-lineal només s'observa per a toleràncies moderades i no és tan notable com en el cas dels problemes escalars. Finalment, els esquemes d'estabilització proposat s'han adaptat a malles de refinament adaptatiu. Concretament, s'han utilitzat malles no-conformes basades en octrees. Utilitzant aquesta configuració, l'eficiència de resoldre un problema altament no-lineal ha estat avaluada de la següent forma. Donada una precisió determinada, el temps computacional requerit per resoldre el problema utilitzant un esquema d'alt ordre ha estat comparat amb el temps necessari per resoldre'l utilitzant un esquema de baix ordre en una malla adaptativa molt més refinada. Addicionalment, també s'ha proposat un estimador de l'error basat en l'operador d'estabilització. El comportament de tots els esquemes proposats anteriorment s'ha avaluat mitjançant varis tests numèrics. Els resultats s'han compilat i comentat en la present tesi.Postprint (published version

Adaptive Multiresolution Methods for the Simulation ofWaves in Excitable Media

We present fully adaptive multiresolution methods for a class of spatially two-dimensional reaction-diffusion systems which describe excitable media and often give rise to the formation of spiral waves. A novel model ingredient is a strongly degenerate diffusion term that controls the degree of spatial coherence and serves as a mechanism for obtaining sharper wave fronts. The multiresolution method is formulated on the basis of two alternative reference schemes, namely a classical finite volume method, and Barkley's approach (Barkley in Phys. D 49:61-70, 1991), which consists in separating the computation of the nonlinear reaction terms from that of the piecewise linear diffusion. The proposed methods are enhanced with local time stepping to attain local adaptivity both in space and time. The computational efficiency and the numerical precision of our methods are assessed. Results illustrate that the fully adaptive methods provide stable approximations and substantial savings in memory storage and CPU time while preserving the accuracy of the discretizations on the corresponding finest uniform gri

Computational Gradient Elasticity and Gradient Plasticity with Adaptive Splines

Classical continuum mechanics theories are largely insufficient in capturing size effects observed in many engineering materials: metals, composites, rocks etc. This is attributed to the absence of a length scale that accounts for microstructural effects inherent in these materials. Enriching the classical theories with an internal length scale solves this problem. One way of doing this, in a theoretically sound manner, is introducing higher order gradient terms in the constitutive relations. In elasticity, introducing a length scale removes the singularity observed at crack tips using the classical theory. In plasticity, it eliminates the spurious mesh sensitivity observed in softening and localisation problems by defining the width of the localisation zone thereby maintaining a well-posed boundary value problem. However, this comes at the cost of more demanding solution techniques. Higher-order continuity is usually required for solving gradient-enhanced continuum theories, a requirement difficult to meet using traditional finite elements. Hermitian finite elements, mixed methods and meshless methods have been developed to meet this requirement, however these methods have their drawbacks in terms of efficiency, robustness or implementational convenience. Isogeometric analysis, which exploits spline-based shape functions, naturally incorporates higher-order continuity, in addition to capturing the exact geometry and expediting the design-through-analysis process. Despite its potentials, it is yet to be fully explored for gradient-enhanced continua. Hence, this thesis develops an isogeometric analysis framework for gradient elasticity and gradient plasticity. The linearity of the gradient elasticity formulation has enabled an operator-split approach so that instead of solving the fourth-order partial differential equation monolithically, a set of two partial differential equations is solved in a staggered manner. A detailed convergence analysis is carried out for the original system and the split set using NURBS and T-splines. Suboptimal convergence rates in the monolithic approach and the limitations of the staggered approach are substantiated. Another advantage of the spline-based approach adopted in this work is the ease with which different orders of interpolation can be achieved. This is useful for consistency, and relevant in gradient plasticity where the local (explicit formulation) or nonlocal (implicit formulation) effective plastic strain needs to be discretised in addition to the displacements. Using different orders of interpolation, both formulations are explored in the second-order and a fourth-order implicit gradient formulation is proposed. Results, corroborated by dispersion analysis, show that all considered models give good regularisation with mesh-independent results. Compared with finite element approaches that use Hermitian shape functions for the plastic multiplier or mixed finite element approaches, isogeometric analysis has the distinct advantage that no interpolation of derivatives is required. In localisation problems, numerical accuracy requires the finite element size employed in simulations to be smaller than the internal length scale. Fine meshes are also needed close to regions of geometrical singularities or high gradients. Maintaining a fine mesh globally can incur high computational cost especially when considering large structures. To achieve this efficiently, selective refinement of the mesh is therefore required. In this context, splines need to be adapted to make them analysis-suitable. Thus, an adaptive isogeometric analysis framework is also developed for gradient elasticity and gradient plasticity. The proposed scheme does not require the mesh size to be smaller than the length scale, even during analysis, until a localisation band develops upon which adaptive refinement is performed. Refinement is based on a multi-level mesh with truncated hierarchical basis functions interacting through an inter-level subdivision operator. Through Bezier extraction, truncation of the bases is simplified by way of matrix multiplication, and an element-wise standard finite element data structure is maintained. In sum, a robust computational framework for engineering analysis is established, combining the flexibility, exact geometry representation, and expedited design-through analysis of isogeometric analysis, size-effect capabilities and mesh-objective results of gradient-enhanced continua, the standard convenient data structure of finite element analysis and the improved efficiency of adaptive hierarchical refinement

Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length

We consider, by means of the variational approximation (VA) and direct numerical simulations of the Gross-Pitaevskii (GP) equation, the dynamics of 2D and 3D condensates with a scattering length containing constant and harmonically varying parts, which can be achieved with an ac magnetic field tuned to the Feshbach resonance. For a rapid time modulation, we develop an approach based on the direct averaging of the GP equation,without using the VA. In the 2D case, both VA and direct simulations, as well as the averaging method, reveal the existence of stable self-confined condensates without an external trap, in agreement with qualitatively similar results recently reported for spatial solitons in nonlinear optics. In the 3D case, the VA again predicts the existence of a stable self-confined condensate without a trap. In this case, direct simulations demonstrate that the stability is limited in time, eventually switching into collapse, even though the constant part of the scattering length is positive (but not too large). Thus a spatially uniform ac magnetic field, resonantly tuned to control the scattering length, may play the role of an effective trap confining the condensate, and sometimes causing its collapse.Comment: 7 figure

Improving multifrontal methods by means of block low-rank representations

Submitted for publication to SIAMMatrices coming from elliptic Partial Differential Equations (PDEs) have been shown to have a low-rank property: well defined off-diagonal blocks of their Schur complements can be approximated by low-rank products. Given a suitable ordering of the matrix which gives to the blocks a geometrical meaning, such approximations can be computed using an SVD or a rank-revealing QR factorization. The resulting representation offers a substantial reduction of the memory requirement and gives efficient ways to perform many of the basic dense algebra operations. Several strategies have been proposed to exploit this property. We propose a low-rank format called Block Low-Rank (BLR), and explain how it can be used to reduce the memory footprint and the complexity of direct solvers for sparse matrices based on the multifrontal method. We present experimental results that show how the BLR format delivers gains that are comparable to those obtained with hierarchical formats such as Hierarchical matrices (H matrices) and Hierarchically Semi-Separable (HSS matrices) but provides much greater flexibility and ease of use which are essential in the context of a general purpose, algebraic solver