DSpace Collection:
http://hdl.handle.net/2117/3918
Tue, 28 Apr 2015 04:03:45 GMT2015-04-28T04:03:45Zwebmaster.bupc@upc.eduUniversitat Politècnica de Catalunya. Servei de Biblioteques i DocumentaciónoJ2 effect and elliptic inclined periodic orbits in the collision three-body problem
http://hdl.handle.net/2117/21117
Title: J2 effect and elliptic inclined periodic orbits in the collision three-body problem
Authors: Barrabes, Esther; Cors Iglesias, Josep Maria; Pinyol, Conxita; Soler Villanueva, Jaume
Abstract: The existence of a new class of inclined periodic orbits of the collision restricted
three{body problem is shown. The symmetric periodic solutions found are perturbations of elliptic
kepler orbits and they exist only for special values of the inclination and are related to the motion
of a satellite around an oblate planet.http://hdl.handle.net/2117/21117Barrabes, Esther; Cors Iglesias, Josep Maria; Pinyol, Conxita; Soler Villanueva, Jaumenocollision restricted three-body problem, periodic orbits, symmetric orbits, critical inclination, continuation methodThe existence of a new class of inclined periodic orbits of the collision restricted
three{body problem is shown. The symmetric periodic solutions found are perturbations of elliptic
kepler orbits and they exist only for special values of the inclination and are related to the motion
of a satellite around an oblate planet.Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates
http://hdl.handle.net/2117/22391
Title: Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates
Authors: Cabré Vilagut, Xavier; Sire, Yannick
Abstract: This is the first of two articles dealing with the equation (-)sv = f (v) in Rn, with s ¿ (0,1), where (-)s stands for the fractional Laplacian — the in¿nitesimal generator of a Lévy process. This equation can be realized as a local linear degenerate elliptic equation in Rn+1+ together with a nonlinear Neumann boundary condition on ¿Rn+1 + =Rn.
In this ¿rst article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of R. These necessary conditions (which will be proven in a follow-up paper to be also suficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian — in the spirit of a result of Modica for the Laplacian. Our proofs are uniform ass ¿1, establishing in the limit the corresponding known results for the Laplacian.
In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.http://hdl.handle.net/2117/22391Cabré Vilagut, Xavier; Sire, YannicknoThis is the first of two articles dealing with the equation (-)sv = f (v) in Rn, with s ¿ (0,1), where (-)s stands for the fractional Laplacian — the in¿nitesimal generator of a Lévy process. This equation can be realized as a local linear degenerate elliptic equation in Rn+1+ together with a nonlinear Neumann boundary condition on ¿Rn+1 + =Rn.
In this ¿rst article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of R. These necessary conditions (which will be proven in a follow-up paper to be also suficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian — in the spirit of a result of Modica for the Laplacian. Our proofs are uniform ass ¿1, establishing in the limit the corresponding known results for the Laplacian.
In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.Unified formalism for the generalized kth-order Hamilton-Jacobi problem
http://hdl.handle.net/2117/27582
Title: Unified formalism for the generalized kth-order Hamilton-Jacobi problem
Authors: Colombo, Leonardo; De León, Manuel; Prieto Martínez, Pedro Daniel; Román Roy, Narciso
Abstract: The geometric formulation of the Hamilton-Jacobi theory enables us to generalize it to systems of higher-order ordinary differential equations. In this work we introduce the unified Lagrangian-Hamiltonian formalism for the geometric Hamilton-Jacobi theory on higher-order autonomous dynamical systems described by regular Lagrangian functions.Fri, 24 Apr 2015 11:51:21 GMThttp://hdl.handle.net/2117/275822015-04-24T11:51:21ZColombo, Leonardo; De León, Manuel; Prieto Martínez, Pedro Daniel; Román Roy, NarcisonoHamilton-Jacobi equation, higher-order Lagrangian and Hamiltonian systems, Skinner-Rusk formalismThe geometric formulation of the Hamilton-Jacobi theory enables us to generalize it to systems of higher-order ordinary differential equations. In this work we introduce the unified Lagrangian-Hamiltonian formalism for the geometric Hamilton-Jacobi theory on higher-order autonomous dynamical systems described by regular Lagrangian functions.An axiomatic characterization of the potential decisiveness index
http://hdl.handle.net/2117/27526
Title: An axiomatic characterization of the potential decisiveness index
Authors: Freixas Bosch, Josep; Pons Navarro, Montserrat
Abstract: Let us consider that somebody is extremely interested in increasing the probability of a proposal to be approved by a certain committee and that to achieve this goal he/she is prepared to pay off one member of the committee. In a situation like this one, and assuming that vote-buying is allowed and free of stigma, which voter should be offered a bribe? The potential decisiveness index for simple games, which measures the effect that ensuring one positive vote produces for the probability of passing the issue at hand, is a good tool with which to acquire the answer. An axiomatic characterization of this index is given in this paper, and its relation to other classical power indices is shown.http://hdl.handle.net/2117/27526Freixas Bosch, Josep; Pons Navarro, MontserratnoGame theory, Potential decisiveness index, Measure for bribes, Axiomatization, Standard power indices, Relationship among several measures, Ordinal equivalence, Voting games, Power, Sucess, Voters, Values, LuckyLet us consider that somebody is extremely interested in increasing the probability of a proposal to be approved by a certain committee and that to achieve this goal he/she is prepared to pay off one member of the committee. In a situation like this one, and assuming that vote-buying is allowed and free of stigma, which voter should be offered a bribe? The potential decisiveness index for simple games, which measures the effect that ensuring one positive vote produces for the probability of passing the issue at hand, is a good tool with which to acquire the answer. An axiomatic characterization of this index is given in this paper, and its relation to other classical power indices is shown.Generalized Clifford-Severi inequality and the volume of irregular varieties
http://hdl.handle.net/2117/27518
Title: Generalized Clifford-Severi inequality and the volume of irregular varieties
Authors: Barja Yáñez, Miguel Ángel
Abstract: We give a sharp lower bound for the self-intersection of a nef line bundle L on an irregular variety X in terms of its continuous global sections and the Albanese dimension of X, which we call the generalized Clifford-Severi inequality. We also extend the result to nef vector bundles and give a slope inequality for fibered irregular varieties. As a by-product we obtain a lower bound for the volume of irregular varieties; when X is of maximal Albanese dimension the bound is vol(X) >= 2n!chi(omega(X)) and it is sharp.Wed, 22 Apr 2015 13:43:14 GMThttp://hdl.handle.net/2117/275182015-04-22T13:43:14ZBarja Yáñez, Miguel ÁngelnoMAXIMAL ALBANESE DIMENSION, ABELIAN-VARIETIES, CANONICAL VOLUME, BICANONICAL MAP, SURFACES, CURVES, FIBRATIONS, FAMILIES, 3-FOLDS, MODULIWe give a sharp lower bound for the self-intersection of a nef line bundle L on an irregular variety X in terms of its continuous global sections and the Albanese dimension of X, which we call the generalized Clifford-Severi inequality. We also extend the result to nef vector bundles and give a slope inequality for fibered irregular varieties. As a by-product we obtain a lower bound for the volume of irregular varieties; when X is of maximal Albanese dimension the bound is vol(X) >= 2n!chi(omega(X)) and it is sharp.Geometric Hamilton-Jacobi theory for higher-order autonomous systems
http://hdl.handle.net/2117/27514
Title: Geometric Hamilton-Jacobi theory for higher-order autonomous systems
Authors: Colombo, Leonardo; De León, Manuel; Prieto Martínez, Pedro Daniel; Román Roy, Narciso
Abstract: The geometric framework for the Hamilton-Jacobi theory is used to study this theory in the background of higher-order mechanical systems, in both the Lagrangian and Hamiltonian formalisms. Thus, we state the corresponding Hamilton-Jacobi equations in these formalisms and apply our results to analyze some particular physical examples.Wed, 22 Apr 2015 12:16:35 GMThttp://hdl.handle.net/2117/275142015-04-22T12:16:35ZColombo, Leonardo; De León, Manuel; Prieto Martínez, Pedro Daniel; Román Roy, NarcisonoHamilton-Jacobi equation, higher-order Lagrangian and Hamiltonian systems, Symplectic geometry, Field-theories, Mechanics, Constraints, Equation, DynamicsThe geometric framework for the Hamilton-Jacobi theory is used to study this theory in the background of higher-order mechanical systems, in both the Lagrangian and Hamiltonian formalisms. Thus, we state the corresponding Hamilton-Jacobi equations in these formalisms and apply our results to analyze some particular physical examples.Isometries on L-2(X) and monotone functions
http://hdl.handle.net/2117/27512
Title: Isometries on L-2(X) and monotone functions
Authors: Boza Rocho, Santiago; Soria, Javier
Abstract: We study necessary and sufficient conditions on a bounded operator T defined on the Hilbert space L-2(X) to be an isometry and show that, under suitable hypotheses, it suffices to restrict T to a smaller class of functions (e.g., if X = R+, to the cone of positive and decreasing functions). We also consider the problem of characterizing the sets Y subset of X for which the orthogonal projection of the operator T on L-2(Y) is also an isometry. Finally, we illustrate our results with several examples involving classical operators on different settings. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimWed, 22 Apr 2015 11:07:17 GMThttp://hdl.handle.net/2117/275122015-04-22T11:07:17ZBoza Rocho, Santiago; Soria, JaviernoIsometries, Hardy operator, monotone functions, Hardy operator, Minus, Decreasing functions, Measure-spaces, Inequalities, Identity, ConeWe study necessary and sufficient conditions on a bounded operator T defined on the Hilbert space L-2(X) to be an isometry and show that, under suitable hypotheses, it suffices to restrict T to a smaller class of functions (e.g., if X = R+, to the cone of positive and decreasing functions). We also consider the problem of characterizing the sets Y subset of X for which the orthogonal projection of the operator T on L-2(Y) is also an isometry. Finally, we illustrate our results with several examples involving classical operators on different settings. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimReal-time monitoring of thermal processes by reduced order modeling
http://hdl.handle.net/2117/27502
Title: Real-time monitoring of thermal processes by reduced order modeling
Authors: Aguado, José Vicente; Huerta, Antonio; Chinesta, Francisco; Cueto Prendes, Elias
Abstract: This work presents a simple technique for real-time monitoring of thermal processes. Real-time simulationbased control of thermal processes is a big challenge because high-fidelity numerical simulations are costly and cannot be used, in general, for real-time decision making. Very often, processes are monitored or controlled with a few measurements at some specific points. Thus, the strategy presented here is centered on fast evaluation of the response only where it is needed. To accomplish this, classical harmonic analysis is combined with recent model reduction techniques. This leads to an advanced harmonic methodology, which solves in real-time the transient heat equation at the monitored point. In order to apply the reciprocity principle, harmonic analysis is used in the space-frequency domain. Then, Proper Generalized Decomposition, a reduced order approach, pre-computes a transfer function able to produce the output response for a given excitation. This transfer function is computed offline and only once. The response at the monitoring point can be recovered performing a computationally inexpensive post-processing step. This last step can be performed online for real-time monitoring of the thermal process. Examples show the applicability of this approach for a wide range of problems ranging from fast temperature evaluation to inverse problems.Wed, 22 Apr 2015 07:31:08 GMThttp://hdl.handle.net/2117/275022015-04-22T07:31:08ZAguado, José Vicente; Huerta, Antonio; Chinesta, Francisco; Cueto Prendes, EliasnoReal-time, Heat Transfer, Monitoring, Model Reduction, Proper Generalized Decomposition, Harmonic AnalysisThis work presents a simple technique for real-time monitoring of thermal processes. Real-time simulationbased control of thermal processes is a big challenge because high-fidelity numerical simulations are costly and cannot be used, in general, for real-time decision making. Very often, processes are monitored or controlled with a few measurements at some specific points. Thus, the strategy presented here is centered on fast evaluation of the response only where it is needed. To accomplish this, classical harmonic analysis is combined with recent model reduction techniques. This leads to an advanced harmonic methodology, which solves in real-time the transient heat equation at the monitored point. In order to apply the reciprocity principle, harmonic analysis is used in the space-frequency domain. Then, Proper Generalized Decomposition, a reduced order approach, pre-computes a transfer function able to produce the output response for a given excitation. This transfer function is computed offline and only once. The response at the monitoring point can be recovered performing a computationally inexpensive post-processing step. This last step can be performed online for real-time monitoring of the thermal process. Examples show the applicability of this approach for a wide range of problems ranging from fast temperature evaluation to inverse problems.A generalization of the Allen–Cahn equation
http://hdl.handle.net/2117/27478
Title: A generalization of the Allen–Cahn equation
Authors: Miranville, Alain; Quintanilla de Latorre, Ramón
Abstract: Our aim in this paper is to study generalizations of the Allen–Cahn equation based on a modification of the Ginzburg–Landau free energy proposed in S. Torabi et al. (2009, A new phase-field model for strongly anisotropic systems. Proc. R. Soc. A, 465, 1337–1359). In particular, the free energy contains an additional term called Willmore regularization. We prove the existence, uniqueness and regularity of solutions, as well as the existence of the global attractor. Furthermore, we study the convergence to the Allen–Cahn equation, when the Willmore regularization goes to zero. We finally study the spatial behaviour of solutions in a semi-infinite cylinder, assuming that such solutions exist.
Description: This is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The version of record: Miranville, A.; Quintanilla, R. A generalization of the Allen–Cahn equation. "IMA Journal of Applied Mathematics", 01 Abril 2015, vol. 80, núm. 2, p. 410-430 is available online at:http://imamat.oxfordjournals.org/content/80/2/410.Tue, 21 Apr 2015 13:31:16 GMThttp://hdl.handle.net/2117/274782015-04-21T13:31:16ZMiranville, Alain; Quintanilla de Latorre, RamónnoAllen–Cahn equation, Willmore regularization, well-posedness, dissipativity, global attractor, spatial behaviourOur aim in this paper is to study generalizations of the Allen–Cahn equation based on a modification of the Ginzburg–Landau free energy proposed in S. Torabi et al. (2009, A new phase-field model for strongly anisotropic systems. Proc. R. Soc. A, 465, 1337–1359). In particular, the free energy contains an additional term called Willmore regularization. We prove the existence, uniqueness and regularity of solutions, as well as the existence of the global attractor. Furthermore, we study the convergence to the Allen–Cahn equation, when the Willmore regularization goes to zero. We finally study the spatial behaviour of solutions in a semi-infinite cylinder, assuming that such solutions exist.Group-theoretic orbit decidability
http://hdl.handle.net/2117/27428
Title: Group-theoretic orbit decidability
Authors: Ventura Capell, Enric
Abstract: A recent collection of papers in the last years have given a renovated interest to the notion of orbit decidability. This is a new quite general algorithmic notion, connecting with several classical results, and closely related to the study of the conjugacy problem for extensions of groups. In the present survey we explain several of the classical results closely
related to this concept, and we explain the main ideas behind the recent connection with the conjugacy problem made by Bogopolski–Martino–Ventura in [2]. All the consequences up to date, published in several other papers by other authors, are also commented and reviewed.Fri, 17 Apr 2015 13:11:15 GMThttp://hdl.handle.net/2117/274282015-04-17T13:11:15ZVentura Capell, EnricnoOrbit decidability, Conjugacy problemA recent collection of papers in the last years have given a renovated interest to the notion of orbit decidability. This is a new quite general algorithmic notion, connecting with several classical results, and closely related to the study of the conjugacy problem for extensions of groups. In the present survey we explain several of the classical results closely
related to this concept, and we explain the main ideas behind the recent connection with the conjugacy problem made by Bogopolski–Martino–Ventura in [2]. All the consequences up to date, published in several other papers by other authors, are also commented and reviewed.Singularities for a fully non-linear elliptic equation in conformal geometry
http://hdl.handle.net/2117/27415
Title: Singularities for a fully non-linear elliptic equation in conformal geometry
Authors: González Nogueras, María del Mar; Mazzieri, LorenzoFri, 17 Apr 2015 09:32:33 GMThttp://hdl.handle.net/2117/274152015-04-17T09:32:33ZGonzález Nogueras, María del Mar; Mazzieri, LorenzonoGeometric Quantization of real polarizations via sheaves
http://hdl.handle.net/2117/27391
Title: Geometric Quantization of real polarizations via sheaves
Authors: Miranda Galcerán, Eva; Presas, Francisco
Abstract: In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular Lagrangian foliation via sheaf cohomology and obtain important new applications in the case of real polarizations. The starting point is the definition of representation spaces due to Kostant. Besides the classical examples of Gelfand-Cetlin systems due to Guillemin and Sternberg [13] very few examples of explicit computations of real polarizations are known. The computation of Geometric Quantization in [13] is based on a theorem due to Śniatycki for fibrations [32] which identifies the representation space with the set of Bohr-Sommerfeld leaves determined by the integral action coordinates.
In this article we check that the associated sheaf cohomology apparatus of Geometric Quantization satisfies Mayer-Vietoris and Künneth formulae. As a consequence, a new short proof of this classical result for fibrations due to Śniatycki is obtained. We also compute Geometric Quantization with respect to any generic regular Lagrangian foliation on a 2-torus and the case of the irrational flow. In the way, we recover some classical results in the computation of foliated cohomology of these polarizations.Thu, 16 Apr 2015 15:07:11 GMThttp://hdl.handle.net/2117/273912015-04-16T15:07:11ZMiranda Galcerán, Eva; Presas, FrancisconoIn this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular Lagrangian foliation via sheaf cohomology and obtain important new applications in the case of real polarizations. The starting point is the definition of representation spaces due to Kostant. Besides the classical examples of Gelfand-Cetlin systems due to Guillemin and Sternberg [13] very few examples of explicit computations of real polarizations are known. The computation of Geometric Quantization in [13] is based on a theorem due to Śniatycki for fibrations [32] which identifies the representation space with the set of Bohr-Sommerfeld leaves determined by the integral action coordinates.
In this article we check that the associated sheaf cohomology apparatus of Geometric Quantization satisfies Mayer-Vietoris and Künneth formulae. As a consequence, a new short proof of this classical result for fibrations due to Śniatycki is obtained. We also compute Geometric Quantization with respect to any generic regular Lagrangian foliation on a 2-torus and the case of the irrational flow. In the way, we recover some classical results in the computation of foliated cohomology of these polarizations.Linearly dependent vectorial decomposition of clutters
http://hdl.handle.net/2117/27361
Title: Linearly dependent vectorial decomposition of clutters
Authors: Martí Farré, Jaume
Abstract: This paper deals with the question of completing a monotone increasing family of
subsets of a finite set
to obtain the linearly dependent subsets of a family of
vectors of a vector space. Specifically, we demonstrate that such vectorial completions
of the family of subsets ¿ exist and, in addition, we show that the minimal
vectorial completions of the family ¿ provide a decomposition of the clutter of the
inclusion-minimal elements of ¿. The computation of such vectorial decomposition
of clutters is also discussed in some cases.Wed, 15 Apr 2015 15:51:42 GMThttp://hdl.handle.net/2117/273612015-04-15T15:51:42ZMartí Farré, JaumenoClutter, Antichain, Hypergraph, Matroid, Decomposition.This paper deals with the question of completing a monotone increasing family of
subsets of a finite set
to obtain the linearly dependent subsets of a family of
vectors of a vector space. Specifically, we demonstrate that such vectorial completions
of the family of subsets ¿ exist and, in addition, we show that the minimal
vectorial completions of the family ¿ provide a decomposition of the clutter of the
inclusion-minimal elements of ¿. The computation of such vectorial decomposition
of clutters is also discussed in some cases.Unstructured and semi-structured hexahedral mesh generation methods
http://hdl.handle.net/2117/27311
Title: Unstructured and semi-structured hexahedral mesh generation methods
Authors: Sarrate Ramos, Josep; Ruiz-Gironés, Eloi; Roca Navarro, Xevi
Abstract: Discretization techniques such as the finite element method, the finite volume method or the discontinuous Galerkin method are the most used simulation techniques in ap- plied sciences and technology. These methods rely on a spatial discretization adapted to the geometry and to the prescribed distribution of element size. Several fast and robust algorithms have been developed to generate triangular and tetrahedral meshes. In these methods local connectivity modifications are a crucial step. Nevertheless, in hexahedral meshes the connectivity modifications propagate through the mesh. In this sense, hexahedral meshes are more constrained and therefore, more difficult to gener- ate. However, in many applications such as boundary layers in computational fluid dy- namics or composite material in structural analysis hexahedral meshes are preferred. In this work we present a survey of developed methods for generating structured and unstructured hexahedral meshes.Tue, 14 Apr 2015 11:16:13 GMThttp://hdl.handle.net/2117/273112015-04-14T11:16:13ZSarrate Ramos, Josep; Ruiz-Gironés, Eloi; Roca Navarro, Xevinomesh generation, quadrilateral mesh, hexahedral mesh, unstructured meshDiscretization techniques such as the finite element method, the finite volume method or the discontinuous Galerkin method are the most used simulation techniques in ap- plied sciences and technology. These methods rely on a spatial discretization adapted to the geometry and to the prescribed distribution of element size. Several fast and robust algorithms have been developed to generate triangular and tetrahedral meshes. In these methods local connectivity modifications are a crucial step. Nevertheless, in hexahedral meshes the connectivity modifications propagate through the mesh. In this sense, hexahedral meshes are more constrained and therefore, more difficult to gener- ate. However, in many applications such as boundary layers in computational fluid dy- namics or composite material in structural analysis hexahedral meshes are preferred. In this work we present a survey of developed methods for generating structured and unstructured hexahedral meshes.Simultaneous untangling and smoothing of quadrilateral and hexahedral meshes using an object-oriented framework
http://hdl.handle.net/2117/27299
Title: Simultaneous untangling and smoothing of quadrilateral and hexahedral meshes using an object-oriented framework
Authors: Ruiz-Gironés, Eloi; Roca Navarro, Xevi; Sarrate Ramos, Josep; Montenegro Armas, Rafael; Escobar Sánchez, José M.
Abstract: In this work, we present a simultaneous untangling and smoothing technique for quadrilateral and hexahedral meshes. The algorithm iteratively improves a quadrilateral or hexahedral mesh by minimizing an objective function defined in terms of a regularized algebraic distortion measure of the elements. We propose several techniques to improve the robustness and the computational efficiency of the optimization algorithm. In addition, we have adopted an object-oriented paradigm to create a common framework to smooth meshes composed by any type of elements, and using different minimization techniques. Finally, we present several examples to show that the proposed technique obtains valid meshes composed by high-quality quadrilaterals and hexahedra, even when the initial meshes contain a large number of tangled elements.Tue, 14 Apr 2015 08:26:24 GMThttp://hdl.handle.net/2117/272992015-04-14T08:26:24ZRuiz-Gironés, Eloi; Roca Navarro, Xevi; Sarrate Ramos, Josep; Montenegro Armas, Rafael; Escobar Sánchez, José M.noMesh generation, Hexahedral mesh, Quality measure, Mesh smoothing, Mesh untangling, Object-oriented framework, TETRAHEDRAL MESH, CONDITION NUMBER, JACOBIAN MATRIX, QUALITY METRICS, ELEMENT MESHES, OPTIMIZATION, ALGORITHM, GENERATION, DECOMPOSITION, IMPROVEMENTIn this work, we present a simultaneous untangling and smoothing technique for quadrilateral and hexahedral meshes. The algorithm iteratively improves a quadrilateral or hexahedral mesh by minimizing an objective function defined in terms of a regularized algebraic distortion measure of the elements. We propose several techniques to improve the robustness and the computational efficiency of the optimization algorithm. In addition, we have adopted an object-oriented paradigm to create a common framework to smooth meshes composed by any type of elements, and using different minimization techniques. Finally, we present several examples to show that the proposed technique obtains valid meshes composed by high-quality quadrilaterals and hexahedra, even when the initial meshes contain a large number of tangled elements.