Relevant bibliographies by topics / Curl-curl problem

Academic literature on the topic 'Curl-curl problem'

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Published: 9 March 2023
Last updated: 11 March 2023

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Journal articles on the topic "Curl-curl problem"

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Zhao, Jikun, and Bei Zhang. "The curl–curl conforming virtual element method for the quad-curl problem." Mathematical Models and Methods in Applied Sciences 31, no. 08 (July 2021): 1659–90. http://dx.doi.org/10.1142/s0218202521500354.

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In this paper, we present the [Formula: see text]-conforming virtual element (VE) method for the quad-curl problem in two dimensions. Based on the idea of de Rham complex, we first construct three families of [Formula: see text]-conforming VEs, of which the simplest one has only one degree of freedom associated to each vertex and each edge in the lowest-order case, respectively. An exact discrete complex is established between the [Formula: see text]-conforming and [Formula: see text]-conforming VEs. We rigorously prove the interpolation error estimates, the stability of discrete bilinear forms, the coercivity and inf–sup condition of the corresponding discrete formulation. We show that the conforming VEs have the optimal convergence. Some numerical examples are given to confirm the theoretical results.
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Mederski, Jarosław. "The Brezis–Nirenberg problem for the curl–curl operator." Journal of Functional Analysis 274, no. 5 (March 2018): 1345–80. http://dx.doi.org/10.1016/j.jfa.2017.12.012.

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Chang, Jumyung, Ruben Partono, Vinicius C. Azevedo, and Christopher Batty. "Curl-Flow." ACM Transactions on Graphics 41, no. 6 (November 30, 2022): 1–21. http://dx.doi.org/10.1145/3550454.3555498.

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We propose to augment standard grid-based fluid solvers with pointwise divergence-free velocity interpolation, thereby ensuring exact incompressibility down to the sub-cell level. Our method takes as input a discretely divergence-free velocity field generated by a staggered grid pressure projection, and first recovers a corresponding discrete vector potential. Instead of solving a costly vector Poisson problem for the potential, we develop a fast parallel sweeping strategy to find a candidate potential and apply a gauge transformation to enforce the Coulomb gauge condition and thereby make it numerically smooth. Interpolating this discrete potential generates a point-wise vector potential whose analytical curl is a pointwise incompressible velocity field. Our method further supports irregular solid geometry through the use of level set-based cut-cells and a novel Curl-Noise-inspired potential ramping procedure that simultaneously offers strictly non-penetrating velocities and incompressibility. Experimental comparisons demonstrate that the vector potential reconstruction procedure at the heart of our approach is consistently faster than prior such reconstruction schemes, especially those that solve vector Poisson problems. Moreover, in exchange for its modest extra cost, our overall Curl-Flow framework produces significantly improved particle trajectories that closely respect irregular obstacles, do not suffer from spurious sources or sinks, and yield superior particle distributions over time.
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Jung, Kyesam, and Gyoungho Lee. "Developing a tutorial to address student difficulties in learning curl: a link between qualitative and mathematical reasoning." Canadian Journal of Physics 90, no. 6 (June 2012): 565–72. http://dx.doi.org/10.1139/p2012-054.

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Many university students have difficulty understanding the concept of curl in our upper-level mechanics courses. This difficulty poses considerable problems when students learn upper-level physics, as the concept of curl is closely related to a wide variety of topics in physics, such as Maxwell’s equations, Faraday’s law, and conservative fields. However, few studies have considered the reasons that students experience difficulty or ways to help students overcome their difficulty grasping the notion of curl. Therefore, in this study, we try to address student difficulty in learning curl. First, we developed a questionnaire to investigate student difficulties in understanding curl and to obtain comments on a tutorial. The questionnaire involved an explanation and a Likert scale form (degrees 0 to 4). We administered a diagnostic test in which we posed two kinds of curl problems. As a result, students who showed qualitative reasoning to solve two types of curl problems were better at solving a curved-line problem than they were at solving a straight-line problem. In contrast, students who showed mathematical reasoning were better at solving the straight-line problem. Based on these results, we developed a tutorial to aid students in using both qualitative and mathematical reasoning. Use of the tutorial enhanced students’ explanations about the curl concept and increased a percentage of correct answers.
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Mederski, Jarosław, Jacopo Schino, and Andrzej Szulkin. "Multiple Solutions to a Nonlinear Curl–Curl Problem in $${\mathbb {R}}^3$$." Archive for Rational Mechanics and Analysis 236, no. 1 (November 21, 2019): 253–88. http://dx.doi.org/10.1007/s00205-019-01469-3.

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AbstractWe look for ground states and bound states $$E:{\mathbb {R}}^3\rightarrow {\mathbb {R}}^3$$E:R3→R3 to the curl–curl problem $$\begin{aligned} \nabla \times (\nabla \times E)= f(x,E) \qquad \text { in } {\mathbb {R}}^3, \end{aligned}$$∇×(∇×E)=f(x,E)inR3,which originates from nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of $$\nabla \times (\nabla \times \cdot )$$∇×(∇×·). The growth of the nonlinearity f is controlled by an N-function $$\Phi :{\mathbb {R}}\rightarrow [0,\infty )$$Φ:R→[0,∞) such that $$\displaystyle \lim _{s\rightarrow 0}\Phi (s)/s^6=\lim _{s\rightarrow +\infty }\Phi (s)/s^6=0$$lims→0Φ(s)/s6=lims→+∞Φ(s)/s6=0. We prove the existence of a ground state, that is, a least energy nontrivial solution, and the existence of infinitely many geometrically distinct bound states. We improve previous results concerning ground states of curl–curl problems. Multiplicity results for our problem have not been studied so far in $${\mathbb {R}}^3$$R3 and in order to do this we construct a suitable critical point theory; it is applicable to a wide class of strongly indefinite problems, including this one and Schrödinger equations.
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BRENNER, SUSANNE C., FENGYAN LI, and LI-YENG SUNG. "A NONCONFORMING PENALTY METHOD FOR A TWO-DIMENSIONAL CURL–CURL PROBLEM." Mathematical Models and Methods in Applied Sciences 19, no. 04 (April 2009): 651–68. http://dx.doi.org/10.1142/s0218202509003565.

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A nonconforming finite element method for a two-dimensional curl–curl problem is studied in this paper. It uses weakly continuous P1 vector fields and penalizes the local divergence. Two consistency terms involving the jumps of the vector fields across element boundaries are also included to ensure the convergence of the scheme. Optimal convergence rates (up to an arbitrary positive ∊) in both the energy norm and the L2 norm are established on graded meshes. This scheme can also be used in the computation of Maxwell eigenvalues without generating spurious eigenmodes. The theoretical results are confirmed by numerical experiments.
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Sun, Jiguang, Qian Zhang, and Zhimin Zhang. "A curl-conforming weak Galerkin method for the quad-curl problem." BIT Numerical Mathematics 59, no. 4 (July 4, 2019): 1093–114. http://dx.doi.org/10.1007/s10543-019-00764-5.

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Aramaki, Junichi. "Variational Problem Involving Operator Curl in a Multiconnected Domain." Chinese Journal of Mathematics 2016 (November 27, 2016): 1–8. http://dx.doi.org/10.1155/2016/2459694.

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We shall study the problem of minimizing a functional involving the curl of vector fields in a three-dimensional, bounded multiconnected domain with prescribed tangential component on the boundary. The paper is an extension of L2 minimization problem of the curl of vector fields. We shall prove the existence and the estimate of minimizers of more general functional which contains Lp norm of the curl of vector fields.
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Mederski, Jarosław, and Andrzej Szulkin. "A Sobolev-Type Inequality for the Curl Operator and Ground States for the Curl–Curl Equation with Critical Sobolev Exponent." Archive for Rational Mechanics and Analysis 241, no. 3 (June 10, 2021): 1815–42. http://dx.doi.org/10.1007/s00205-021-01684-x.

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AbstractLet $$\Omega \subset \mathbb {R}^3$$ Ω ⊂ R 3 be a Lipschitz domain and let $$S_\mathrm {curl}(\Omega )$$ S curl ( Ω ) be the largest constant such that $$\begin{aligned} \int _{\mathbb {R}^3}|\nabla \times u|^2\, \mathrm{d}x\ge S_{\mathrm {curl}}(\Omega ) \inf _{\begin{array}{c} w\in W_0^6(\mathrm {curl};\mathbb {R}^3)\\ \nabla \times w=0 \end{array}}\Big (\int _{\mathbb {R}^3}|u+w|^6\,\mathrm{d}x\Big )^{\frac{1}{3}} \end{aligned}$$ ∫ R 3 | ∇ × u | 2 d x ≥ S curl ( Ω ) inf w ∈ W 0 6 ( curl ; R 3 ) ∇ × w = 0 ( ∫ R 3 | u + w | 6 d x ) 1 3 for any u in $$W_0^6(\mathrm {curl};\Omega )\subset W_0^6(\mathrm {curl};\mathbb {R}^3)$$ W 0 6 ( curl ; Ω ) ⊂ W 0 6 ( curl ; R 3 ) , where $$W_0^6(\mathrm {curl};\Omega )$$ W 0 6 ( curl ; Ω ) is the closure of $$\mathcal {C}_0^{\infty }(\Omega ,\mathbb {R}^3)$$ C 0 ∞ ( Ω , R 3 ) in $$\{u\in L^6(\Omega ,\mathbb {R}^3): \nabla \times u\in L^2(\Omega ,\mathbb {R}^3)\}$$ { u ∈ L 6 ( Ω , R 3 ) : ∇ × u ∈ L 2 ( Ω , R 3 ) } with respect to the norm $$(|u|_6^2+|\nabla \times u|_2^2)^{1/2}$$ ( | u | 6 2 + | ∇ × u | 2 2 ) 1 / 2 . We show that $$S_{\mathrm {curl}}(\Omega )$$ S curl ( Ω ) is strictly larger than the classical Sobolev constant S in $$\mathbb {R}^3$$ R 3 . Moreover, $$S_{\mathrm {curl}}(\Omega )$$ S curl ( Ω ) is independent of $$\Omega $$ Ω and is attained by a ground state solution to the curl–curl problem $$\begin{aligned} \nabla \times (\nabla \times u) = |u|^4u \end{aligned}$$ ∇ × ( ∇ × u ) = | u | 4 u if $$\Omega =\mathbb {R}^3$$ Ω = R 3 . With the aid of these results we also investigate ground states of the Brezis–Nirenberg-type problem for the curl–curl operator in a bounded domain $$\Omega $$ Ω $$\begin{aligned} \nabla \times (\nabla \times u) +\lambda u = |u|^4u\quad \hbox {in }\Omega , \end{aligned}$$ ∇ × ( ∇ × u ) + λ u = | u | 4 u in Ω , with the so-called metallic boundary condition $$\nu \times u=0$$ ν × u = 0 on $$\partial \Omega $$ ∂ Ω , where $$\nu $$ ν is the exterior normal to $$\partial \Omega $$ ∂ Ω .
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Nicaise, Serge. "Singularities of the quad curl problem." Journal of Differential Equations 264, no. 8 (April 2018): 5025–69. http://dx.doi.org/10.1016/j.jde.2017.12.032.

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Dissertations / Theses on the topic "Curl-curl problem"

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BERNINI, FEDERICO. "Different approaches in Critical Point Theory for entire Schrödinger equations and one for curl-curl problems." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2022. http://hdl.handle.net/10281/378952.

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Scopo di questa tesi è mostrare i risultati ottenuti per tre equazioni differenziali alle derivate parziali ellittiche di tipo Schrödinger. Queste equazioni, sebbene condividano la particolarità di essere definite in tutto lo spazio, sono state affrontate con diversi metodi, e per ognuna è stato fornito un risultato di esistenza di soluzioni. Rimarchiamo che l'ultima equazione ha un forte legame con le equazioni di Maxwell. Problema 1) Consideriamo un'equazione di tipo Schrödinger, con potenziale di tipo convolutivo ed una nonlinearità perturbata e pesata derivante dalla fisica quantistica con gravitazione Newtoniana. Se consideriamo questa equazione definita in tutto lo spazio R2, otterremo un potenziale di tipo logaritmico, che rende l'analisi più delicata, in quanto il funzionale associato non è ben definito. Va dunque introdotto un opportuno setting variazionale per dimostrare la buona positura del problema. Successivamente, per gestire la perturbazione utilizziamo la tecnica perturbativa della teoria dei punti critici. Assumendo opportune ipotesi sulla funzione peso, si dimostra l'esistenza di soluzioni locali e globali. Problema 2) La seconda equazione è un'equazione di tipo Choquard governata da un operatore semirelativistico di Schrödinger, dove il potenziale ha una parte singolare ed è presente una nonlinearità generale, definita in tutto lo spazio RN. Grazie alla rappresentazione mediante trasformata di Fourier dell'operatore semirelativistico, si può dimostrare che la norma generata dalla forma quadratica associata al problema è equivalente alla norma standard. Grazie ad un risultato astratto, si prova prima l'esistenza di una successione di Cerami e successivamente la sua limitatezza. Adattando un argomento di decomposizione per successioni di Palais-Smale, si dimostra poi la convergenza di questa successione ad un punto critico non banale. Infine, viene fornita una quasi-caratterizzazione per l'esistenza di soluzioni di tipo ground-state (i.e. soluzioni corrispondenti al livello di energia minima del sistema). Per queste soluzioni, è fornito anche un risultato di compattezza rispetto al termine singolare. Problema 3) Viene fornito un Teorema astratto di tipo linking infinito-dimensionale che permette lo studio di problemi fortemente indefiniti (cioè il punto origine appartiene ad un gap spettrale dell'operatore) e con nonlinearità generali di segno variabile. Come applicazione, questo Teorema viene applicato ad un'equazione di tipo Schrödinger fortemente indefinita con potenziale singolare e nonlinearità a segno variabile, definita in tutto lo spazio RN. Per questa equazione viene dimostrata l'esistenza di una soluzione non banale. Sfruttando un risultato di equivalenza, viene fornita l'esistenza di una soluzione nonbanale anche per un'equazione di tipo curl-curl: questo tipo di equazioni sono strettamente legate alle equazioni di Maxwell.
The purpose of this thesis is to show the results obtained for three Schrödinger type elliptic partial differential equations. These equations, although sharing the feature of being entire, i.e. defined in the whole the space, have been approached with different methods, and for each a result of the existence of solutions has been provided. We emphasize that the last equation has a strong connection with Maxwell's equations. Problem 1) Let us consider a Schrödinger type equation, with convolutive potential and a perturbed and weighted nonlinearity copuling from quantum physics with Newtonian gravitation. If we consider this equation defined in all the space R2, we will obtain a logarithmic type potential, which makes the analysis more delicate, since the associated functional is not well defined. Therefore, a suitable variational setting must be introduced to show the well-posedness of the problem. Next, to manage the perturbation we use the perturbation technique of the critical point theory. Assuming suitable hypotheses on the weight function, the existence of local and global solutions is proved. Problem 2) The second equation is a Choquard type equation driven by a semirelativistic Schrödinger operator, defined in the whole space RN, where the potential has a singular part and a general nonlinearity is considered. Using the Fourier transform representation of the semirelativistic operator, it can be shown that the norm generated by the quadratic form associated with the problem is equivalent to the standard one. Thanks to an abstract result, we first prove the existence of a Cerami-sequence and then its boundedness. By adapting a Palais-Smale-sequence decomposition argument, the strong convergence of this sequence to a non-trivial critical point is then showed. Finally, an almost-characterization criterion is provided for the existence of ground-state solutions (i.e. solutions corresponding to the minimum energy level of the system). For these solutions, a compactness result is also given with respect to the singular term. Problem 3) An abstract infinite-dimensional linking-type Theorem is provided, which allows the study of strongly indefinite problems (i.e. the origin belongs to a spectral gap of the operator) and with general sign-changing nonlinearities. As an application, this Theorem is applied to a strongly indefinite Schrödinger equation with singular potential and sign-changing nonlinearity, defined in the whole space RN. For this equation the existence of a non-trivial solution is proved. By exploiting an equivalence result, the existence of a non-trivial solution is also provided for a curl-curl equation: this type of equations are closely related to Maxwell's equations.
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Bonazzoli, Marcella. "Méthodes d'ordre élevé et méthodes de décomposition de domaine efficaces pour les équations de Maxwell en régime harmonique." Thesis, Université Côte d'Azur (ComUE), 2017. http://www.theses.fr/2017AZUR4067/document.

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Les équations de Maxwell en régime harmonique comportent plusieurs difficultés lorsque la fréquence est élevée. On peut notamment citer le fait que leur formulation variationnelle n’est pas définie positive et l’effet de pollution qui oblige à utiliser des maillages très fins, ce qui rend problématique la construction de solveurs itératifs. Nous proposons une stratégie de solution précise et rapide, qui associe une discrétisation par des éléments finis d’ordre élevé à des préconditionneurs de type décomposition de domaine. La conception, l’implémentation et l’analyse des deux méthodes sont assez difficiles pour les équations de Maxwell. Les éléments finis adaptés à l’approximation du champ électrique sont les éléments finis H(rot)-conformes ou d’arête. Ici nous revisitons les degrés de liberté classiques définis par Nédélec, afin d’obtenir une expression plus pratique par rapport aux fonctions de base d’ordre élevé choisies. De plus, nous proposons une technique pour restaurer la dualité entre les fonctions de base et les degrés de liberté. Nous décrivons explicitement une stratégie d’implémentation qui a été appliquée dans le langage open source FreeFem++. Ensuite, nous nous concentrons sur les techniques de préconditionnement du système linéaire résultant de la discrétisation par éléments finis. Nous commençons par la validation numérique d’un préconditionneur à un niveau, de type Schwarz avec recouvrement, avec des conditions de transmission d’impédance entre les sous-domaines. Enfin, nous étudions comment des préconditionneurs à deux niveaux, analysés récemment pour l’équation de Helmholtz, se comportent pour les équations de Maxwell, des points de vue théorique et numérique. Nous appliquons ces méthodes à un problème à grande échelle qui découle de la modélisation d’un système d’imagerie micro-onde, pour la détection et le suivi des accidents vasculaires cérébraux. La précision et la vitesse de calcul sont essentielles dans cette application
The time-harmonic Maxwell’s equations present several difficulties when the frequency is large, such as the sign-indefiniteness of the variational formulation, the pollution effect and the problematic construction of iterative solvers. We propose a precise and efficient solution strategy that couples high order finite element (FE) discretizations with domain decomposition (DD) preconditioners. High order FE methods make it possible for a given precision to reduce significantly the number of unknowns of the linear system to be solved. DD methods are then used as preconditioners for the iterative solver: the problem defined on the global domain is decomposed into smaller problems on subdomains, which can be solved concurrently and using robust direct solvers. The design, implementation and analysis of both these methods are particularly challenging for Maxwell’s equations. FEs suited for the approximation of the electric field are the curl-conforming or edge finite elements. Here, we revisit the classical degrees of freedom (dofs) defined by Nédélec to obtain a new more friendly expression in terms of the chosen high order basis functions. Moreover, we propose a general technique to restore duality between dofs and basis functions. We explicitly describe an implementation strategy, which we embedded in the open source language FreeFem++. Then we focus on the preconditioning of the linear system, starting with a numerical validation of a one-level overlapping Schwarz preconditioner, with impedance transmission conditions between subdomains. Finally, we investigate how two-level preconditioners recently analyzed for the Helmholtz equation work in the Maxwell case, both from the theoretical and numerical points of view. We apply these methods to the large scale problem arising from the modeling of a microwave imaging system, for the detection and monitoring of brain strokes. In this application accuracy and computing speed are indeed of paramount importance
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"A posteriori error estimate for H(curl)-elliptic problem by staggered DG method." 2013. http://library.cuhk.edu.hk/record=b5884345.

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Yuen, Man Chun.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2013.
Includes bibliographical references (leaves 58-[60]).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
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Li, Wei-De, and 李威德. "Eigenvalue Problems of Discrete Curl Operators on Various Lattices for Simulating Three Dimensional Photonic Crystals." Thesis, 2017. http://ndltd.ncl.edu.tw/handle/um39q2.

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博士
國立清華大學
數學系所
106
Photonic crystal is one of the most important materials in optoelectronics technology, it is made up of periodic dielectrics and imitates the arrangement of atoms in the crystal. The basic optical property of a photonic crystal is band gap, it is necessary to use the numerical method when computing the band gap. The propagation behavior of electromagnetic waves in photonic crystals is governed by Maxwell's equations, applying the time harmonic assumption, the governing equations will become a time-independent frequency domain problem. There are many numerical methods, after comparison, we believe that Yee's scheme is the best discrete method. So far, this method can only be applied to simple cubic lattice and face-centered cubic lattice, but there are 14 Bravais lattices in three-dimensional space. The first important work of this dissertation is to extend this method to all of the Bravais lattices. Using the Yee's scheme to discretize the Maxwell's equations, we can get a general eigenvalue problem, and we will analyze this general eigenvalue problem. The second important task of this article is to find the eigen-decomposition of the discrete curl operator, after a series of complicated calculations, we find that all the lattices can be summed up into two kinds of decomposition. We are interested in finding the several few smallest real eigenvalues, but the large dimension of null space is 1/3 of all, this seriously affected the convergence of calculation, we use a technique which is called nullspace-free method to avoid this trouble. But this technique transforms the sparse matrix in our problem to a dense matrix, fortunately, the eigenvectors we found before are related to discrete Fourier transformation. The efficiency of calculation has been significantly improved by using the fast Fourier transformation. Finally, we calculate the band structures of photonic crystals on various lattices, and implement high performance calculations on the GPU.
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Books on the topic "Curl-curl problem"

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Nicolaides, Roy A. Direct discretization of planar div-curl problems. [Washington, DC: National Aeronautics and Space Administration, 1989.

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Nicolaides, R. A. Direct discretization of planar div-curl problems. Hampton, Va: ICASE, 1989.

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Book chapters on the topic "Curl-curl problem"

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Ayuso de Dios, Blanca, Ralf Hiptmair, and Cecilia Pagliantini. "Auxiliary Space Preconditioners for a DG Discretization of H(curl; Ω)-Elliptic Problem on Hexahedral Meshes." In Lecture Notes in Computational Science and Engineering, 223–31. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-93873-8_20.

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Gerritsma, Marc, Varun Jain, Yi Zhang, and Artur Palha. "Algebraic Dual Polynomials for the Equivalence of Curl-Curl Problems." In Lecture Notes in Computational Science and Engineering, 307–20. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-30705-9_27.

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Casagrande, Raffael, Christoph Winkelmann, Ralf Hiptmair, and Joerg Ostrowski. "DG Treatment of Non-conforming Interfaces in 3D Curl-Curl Problems." In Scientific Computing in Electrical Engineering, 53–61. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-30399-4_6.

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Kolev, Tzanio V., and Panayot S. Vassilevski. "Auxiliary Space AMG for H(curl) Problems." In Lecture Notes in Computational Science and Engineering, 147–54. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008. http://dx.doi.org/10.1007/978-3-540-75199-1_13.

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Xu, Jinchao, and Yunrong Zhu. "Robust Preconditioner for H(curl) Interface Problems." In Lecture Notes in Computational Science and Engineering, 173–80. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-11304-8_18.

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Toselli, Andrea, and Olof B. Widlund. "Problems in H (div ; ω) and H (curl ; ω)." In Springer Series in Computational Mathematics, 271–309. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005. http://dx.doi.org/10.1007/3-540-26662-3_10.

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Mazza, Mariarosa, Carla Manni, and Hendrik Speleers. "Spectral Analysis of Isogeometric Discretizations of 2D Curl-Div Problems with General Geometry." In Lecture Notes in Computational Science and Engineering, 251–62. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-39647-3_19.

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Dohrmann, Clark R., and Olof B. Widlund. "Some Recent Tools and a BDDC Algorithm for 3D Problems in H(curl)." In Lecture Notes in Computational Science and Engineering, 15–25. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-35275-1_2.

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Wang, Junxian, Shi Shu, and Liuqiang Zhong. "Efficient Parallel Preconditioners for High-Order Finite Element Discretizations of H(grad) and H(curl) Problems." In Lecture Notes in Computational Science and Engineering, 325–32. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-11304-8_37.

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Glatzmaier, Gary A. "Spatial Discretizations." In Introduction to Modeling Convection in Planets and Stars. Princeton University Press, 2013. http://dx.doi.org/10.23943/princeton/9780691141725.003.0009.

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This chapter considers two ways of employing a spatial resolution that varies with position within a finite-difference method: using a nonuniform grid and mapping to a new coordinate variable. It first provides an overview of nonuniform grids before discussing coordinate mapping as an alternative way of achieving spatial discretization. It then describes an approach for treating both the vertical and horizontal directions with simple finite-difference methods: defining a streamfunction, which automatically satisfies mass conservation, and solving for vorticity via the curl of the momentum conservation equation. It also explains the use of the Chebyshev–Fourier method to simulate the convection or gravity wave problem by employing spectral methods in both the horizontal and vertical directions. Finally, it looks at the basic ideas and some issues that need to be addressed with respect to parallel processing as well as choices that need to be made when designing a parallel code.
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Conference papers on the topic "Curl-curl problem"

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Xia, Z. C., and F. Ren. "An Investigation of Wall Curl Reduction Through Post-Stretch Forming." In ASME 2004 International Mechanical Engineering Congress and Exposition. ASMEDC, 2004. http://dx.doi.org/10.1115/imece2004-60604.

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When sheet metals go through drawbeads or die corners, stress differentials are generated across metal thickness. The draw wall will curl up upon release of stamping tools, resulting in so-called wall curl. It is a serious problem in the deep drawings of U-channel type of structures such as rails. Numerical modeling is conducted to investigate a post-stretch forming process for wall curl reduction. In this process a set of lockbeads in the binder activates just before the end of punch stroke, locking the remaining blank in the binder. The continuation of the punch stroke then creates a final increment of pure stretch. It is most effective for deep drawings of U-channel type of structures such as rails. This technique is also known as “post-stretch” or “shape set” in the automotive industry and in the literature. Finite element simulations for a straight channel are conducted in order to understand the wall curl reduction mechanism of the process and to determine its effectiveness. After an examination of deformation profile after drawing and wall curl as a result of springback, various magnitude of post-stretch amount is modeled and their deformation history is analyzed. It is found that a post-stretch strain around 2% almost completely eliminates wall curl. CAE investigations demonstrate that the technique is equally applicable to more complex 3D channel, where a step channel is examined. The effectiveness of this concept is demonstrated by laboratory experiment on the forming of a U-channel. Various implementation techniques for the process in an industrial environment are also suggested, together with a discussion on the associated benefits and costs for production use.
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Liu, Li, and Jyhwen Wang. "Springback Analysis for Metal-Polymer-Metal Laminates." In ASME 2003 International Mechanical Engineering Congress and Exposition. ASMEDC, 2003. http://dx.doi.org/10.1115/imece2003-42550.

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Metal-polymer-metal laminate is an emerging material that has many potential applications. The laminated structure consists of two outer layers of sheet metal and a polymeric center core. The material offers an excellent sound deadening properties and is being introduced to applications where noise reduction is desired. Part manufacturing for laminates involves converting a flat sheet into a deformed body. Springback has been a major concern in shape control. While bending of a single layered sheet metal does not exhibit significant sidewall curl, the problem is pronounced in bending laminates. This paper presents an analytical approach to predict springback and sidewall curl of laminates due to simple bending. Based on the models, springback factor Ks is calculated. It is shown that the prediction is in good agreement with the published experimental data.
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Yang, Puyi, and Hamidreza Najafi. "A Comparative Study of the Influence of Different Wake Models on Wind Farm Layout Optimization." In ASME 2021 15th International Conference on Energy Sustainability collocated with the ASME 2021 Heat Transfer Summer Conference. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/es2021-63911.

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Abstract The accuracy of analytical wake models applied in wind farm layout optimization (WFLO) problems plays a vital role in the present era that the high-fidelity methods such as LES and RANS are still not able to handle an optimization problem for large wind farms. Based on a verity of analytical wake models developed in the past decades, FLOw Redirection and Induction in Steady State (FLORIS) has been published as a tool integrated several widely used wake models and the expansions for them. This paper compares four wake models selected from FLORIS by applying three classical WFLO scenarios. The results illustrate that the Jensen wake model is the fastest one but the defect of underestimation of velocity deficit is obvious. The Multi Zone model needs to be applied additional tunning on the parameters inside the model to fit specific wind turbines. The Gaussian-Curl wake model as an advanced expansion of the Gaussian wake model does not perform an observable improvement in the current study that the yaw control is not included. The default Gaussian wake model is recommended to be used in the WFLO projects which implemented under the FLROIS framework and has similar wind conditions with the present work.
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Chen, Kuangxu, and Chunlei Liang. "A High-Order Spectral Difference Solver for 2D Ideal MHD Equations With Constrained Transport." In ASME 2021 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/imece2021-73359.

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Abstract When using the discontinuous Galerkin or Spectral Difference (SD) method to discretize ideal magnetohydrodynamic (MHD) equations, it is challenging to satisfy the divergence-free constraint for the magnetic field over long-period time integration. To tackle this challenge, the SD method is integrated with an unstaggered Constrained Transport approach (SDCT). In addition to solving the two-dimensional ideal MHD equations, one more equation describing the transport of the magnetic potential is introduced. After each time step, the magnetic field will be updated by computing the curl of the magnetic potential. This strategy preserves ∇ · B = 0 exactly by construction in the discrete sense. Meanwhile, the additional computational cost is only approximately 20% more than that without the constrained transport. Moreover, the inclusion of the constrained transport does not obstruct the implementation of the artificial viscosity for shock capturing. Several well-known benchmark test cases are studied in this paper using the SDCT method. In the magnetic field loop advection test, the proposed SDCT avoids spurious growth of magnetic energy, and the numerical dissipation is shown to decrease when increasing the polynomial degree while maintaining the total degrees of freedom. In the propagation of Alfvén wave problem, the high-order accuracies of the SDCT method are verified. In the Orzag-Tang vortex problem, the predicted pressure distribution and density contours match well with those in the reference [1]. Meanwhile, a mesh convergence study shows that the SDCT method equipped with the artificial viscosity terms can produce converged results even in the vicinity of shocks.
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Choudhary, Aniruddha, Christopher J. Roy, Jean-François Dietiker, Mehrdad Shahnam, and Rahul Garg. "Code Verification for Multiphase Flows Using the Method of Manufactured Solutions." In ASME 2014 4th Joint US-European Fluids Engineering Division Summer Meeting collocated with the ASME 2014 12th International Conference on Nanochannels, Microchannels, and Minichannels. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/fedsm2014-21608.

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Code verification is the process of ensuring, to the degree possible, that there are no algorithm deficiencies and coding mistakes (bugs) in a computational fluid dynamics (CFD) code. In order to perform code verification, the Method of Manufactured Solutions (MMS) is a rigorous technique that can be used in the absence of exact solution to the problem. This work addresses major aspects of performing code verification for multiphase flow codes using the open-source, multiphase flow code MFIX which employs a staggered-grid and a modified SIMPLE-based algorithm. Code verification is performed on 2D and 3D, uniform and stretched meshes for incompressible, steady and unsteady, single-phase and two-phase flows using the two-fluid model of MFIX. Currently, the algebraic gas-solid exchange terms are neglected as these can be tested via unit-testing. The no-slip wall, free-slip wall, and pressure outflow boundary conditions are verified for 2D and 3D flows. A newly-developed curl-based manufactured solution for 3D divergence free flows is introduced. Temporal order of accuracy during unsteady calculations is also assessed. Techniques are introduced to generate manufactured solutions that satisfy the divergence-free constraint during the verification of the incompressible governing equations. Manufactured solutions with constraints due to boundary conditions as well as due to divergence-free flow are derived in order to verify the boundary conditions. Use of staggered grid and SIMPLE-based algorithm for numerical computations in MFIX requires specific issues to be addressed while performing MMS-based code verification. Lessons learned during this code verification exercise are discussed.
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Grosu, Corina, and Marta Grosu. "ENVIRONMENTAL MODELING THROUGH MATH GAME." In eLSE 2017. Carol I National Defence University Publishing House, 2017. http://dx.doi.org/10.12753/2066-026x-17-045.

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Forming adequate habits in the process of teaching the fundamental sciences may improve the sustainable contribution to a responsible environmental politic. Our present paper is focusing on such a problem in connection to the analysis of the pollution contaminating groundwater resources. Since mathematics plays a major role in the environmental modeling, the understanding and recognition of abstract notions in well specified problems- a theme present in most of our papers- can be successfully obtained by merging them in an e-learning designed game. The conceptual models on which the process of analysis and decision takes plays rely, in this case, on the use of certain equations which govern a fluid flow through different media. The students should be capable of understanding the extent to which the model is suitable in characterizing polluted aquifer systems along with its limitations, thus they will have to acquire, through their first year study at Politehnica University, notions belonging to different parts of mathematics. Our present game is concentrated over some of these notions which are required by Darcy's law along with the continuity equation. These in turn reside on the understanding of the difference between partial derivatives and total derivatives. Key concepts like curl or divergence of the flow of a fluid play an important role both in the theoretical models as well as in the game's strategy. The starting point for our game is the discovery, by our heroin, of an underground polluted river spreading its tainted waters in the nearby lake. In her journey to the source of pollution, our heroin faces natural and artificial enemies: alligators in quest for food, poisonous flying fishes, but also mischievous guards of pollutant reservoirs. Mastering game' levels and math' levels, in a precisely defined succession, the heroin follows the path which leads from the shore of the lake up to the mountain where the source of pollution (a well hidden lab) lies.
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Suyitno, L. Katgerman, and A. Burghardt. "Determination of Boundary Conditions During the Start-Up of DC Casting." In ASME 2002 International Mechanical Engineering Congress and Exposition. ASMEDC, 2002. http://dx.doi.org/10.1115/imece2002-33830.

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The start-up phase of DC (direct chill) casting of aluminium alloys is a critical stage during which unwanted phenomena can occur, for example: butt curl, ingot cracking, and re-melting. To avoid these problems, simulation of the DC casting process during this phase is very important. However, the simulation task is not an easy job, because the boundary conditions to be applied are not easily defined and established. To define the boundary condition inverse methods are used in combination with experimental data. The aim of the simulations is intended to predict butt curl and butt temperatures. By application of inversely determined boundary conditions, results can be obtained that correlate well to measured data, even for different casting practices.
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Shurina, Ella P., and Ekaterina I. Mikhaylova. "Modified multiscale discontinuous Galerkin method in the function space H(curl)." In 2016 13th International Scientific-Technical Conference on Actual Problems of Electronics Instrument Engineering (APEIE). IEEE, 2016. http://dx.doi.org/10.1109/apeie.2016.7806499.

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Shurina, E. P., and E. I. Mikhaylova. "Modified multiscale discontinuous Galerkin method in the function space H(curl)." In 2016 13th International Scientific-Technical Conference on Actual Problems of Electronics Instrument Engineering (APEIE). IEEE, 2016. http://dx.doi.org/10.1109/apeie.2016.7806963.

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Li, Yan Lin, Sheng Sun, Qi I. Dai, and Weng Cho Chew. "Vectorial solution to double curl equation with generalized coulomb gauge for magneto static problems." In 2015 IEEE International Conference on Computational Electromagnetics (ICCEM). IEEE, 2015. http://dx.doi.org/10.1109/compem.2015.7052658.

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Reports on the topic "Curl-curl problem"

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Kolev, T., and P. Vassilevski. Parallel H1-based auxiliary space AMG solver for H(curl) problems. Office of Scientific and Technical Information (OSTI), June 2006. http://dx.doi.org/10.2172/897951.

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Kolev, T., and P. Vassilevski. Parallel eigensolver for H(curl) problems using H1-auxiliary space AMG preconditioning. Office of Scientific and Technical Information (OSTI), November 2006. http://dx.doi.org/10.2172/900179.

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Kolev, T., and P. Vassilevski. Some experience with a H1-based auxiliary space AMG for H(curl)-problems. Office of Scientific and Technical Information (OSTI), May 2006. http://dx.doi.org/10.2172/900464.

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Lapidot, Moshe, Linda Hanley-Bowdoin, Jane E. Polston, and Moshe Reuveni. Geminivirus-resistant Tomato Plants: Combining Transgenic and Conventional Strategies for Multi-viral Resistance. United States Department of Agriculture, December 2010. http://dx.doi.org/10.32747/2010.7592639.bard.

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Begomoviruses, which constitute one genus of the Geminiviridae family, are single-stranded DNA viruses that infect many dicotyledonous crops important to large agricultural industries as well as to subsistence growers. Although all begomoviruses are transmitted by whiteflies (Bemisia tabaci), they have proven difficult to manage even with heavy insecticide applications. The begomovirus, Tomato yellow leaf curl virus (TYLCV), has been a problem in tomato production in Israel since the 1950s and in the United States since 1997. Approximately 89 begomoviruses have now been reported to infect tomato. Crop losses due to begomoviruses such as TYLCV and Tomato mottle virus (ToMoV), are limiting factors in tomato cultivation in Israel, the U.S., and many tomato-growing regions throughout the world. To overcome these limitations, we proposed a two-step strategy that combines transgenic and conventional resistance in order to develop tomato plants that are resistant to multiple begomoviruses. In the first step, we have developed transgenic tomato plants expressing trans-dominant interfering mutants Rep and C3 from TYLCV and ToMoV, and tested whether these plants are resistant to infection by these two viruses. In the second step we have tested whether pyramiding transgenic and conventional resistance is superior to either strategy alone. The specific objectives of the proposal were: 1. Design and test trans-dominant interfering constructs for TYLCV and ToMoV Rep and C3 in transient replication interference assays. 2. Generate and test transgenic tomato plants expressing mutant Rep and C3 in resistance assays. 3. Generate and test conventional resistant lines that also express mutant Rep and C3. Two viral replication interfering constructs, expressing the trans-dominant interfering mutants Rep and C3, were designed and constructed during this project. One construct, pNSB1630 was based on TYLCV sequences and the other, pNSB1682, based on ToMoV sequences. The TYLCV transformation construct was tested in a protoplasts replication assay, and was found to inhibit TYLCV replication. The ToMoV transformation construct is yet to be tested in a protoplast assay. Both transformation vectors, pNSB1630 and pNSB1682, were used to transform four different tomato lines, and generate transgenic plants. The tomato lines used for transformation were: FL7613, MM, TY172, TY199. FL7613 and MM are susceptible to both TYLCV and ToMoV. TY172 and TY199 are breeding lines developed at Volcani Center. TY172 is resistant to TYLCV but susceptible to ToMoV, while TY199 is resistant to both TYLCV and ToMoV. When transgenic T1 plants expressing the pNSB1630 constructed were screened for TYLCV resistance, it was found that these plants showed very low level of TYLCV resistance, if any. However, some of these lines showed high level of resistance to ToMoV. Only five transgenic T1 lines expressing the pNSB1682 construct were tested (so far) for resistance to ToMoV. It was found that all five lines express very high level of resistance to ToMoV. Although we haven’t finished (yet) the screen of all the transgenic lines, it is already clear that we were able to successfully combine genetic resistance for TYLCV with transgenic resistance to ToMoV.
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Kalchev, Delyan, Panayot Vassilevski, and Umberto Villa. On ParELAG's Parallel Element-based Algebraic Multigrid and its MFEM Miniapps for H(curl) and H(div) Problems: a report including lowest and next to the lowest order numerical results. Office of Scientific and Technical Information (OSTI), July 2021. http://dx.doi.org/10.2172/1807757.

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Ghanim, Murad, Joe Cicero, Judith K. Brown, and Henryk Czosnek. Dissection of Whitefly-geminivirus Interactions at the Transcriptomic, Proteomic and Cellular Levels. United States Department of Agriculture, February 2010. http://dx.doi.org/10.32747/2010.7592654.bard.

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Our project focuses on gene expression and proteomics of the whitefly Bemisia tabaci (Gennadius) species complex in relation to the internal anatomy and localization of expressed genes and virions in the whitefly vector, which poses a major constraint to vegetable and fiber production in Israel and the USA. While many biological parameters are known for begomovirus transmission, nothing is known about vector proteins involved in the specific interactions between begomoviruses and their whitefly vectors. Identifying such proteins is expected to lead to the design of novel control methods that interfere with whitefly-mediated begomovirus transmission. The project objectives were to: 1) Perform gene expression analyses using microarrays to study the response of whiteflies (B, Q and A biotypes) to the acquisition of begomoviruses (Tomato yellow leaf curl (TYLCV) and Squash leaf curl (SLCV). 2) Construct a whitefly proteome from whole whiteflies and dissected organs after begomovirus acquisition. 3) Validate gene expression by q-RTPCR and sub-cellular localization of candidate ESTs identified in microarray and proteomic analyses. 4) Verify functionality of candidate ESTs using an RNAi approach, and to link these datasets to overall functional whitefly anatomical studies. During the first and second years biological experiments with TYLCV and SLCV acquisition and transmission were completed to verify the suitable parameters for sample collection for microarray experiments. The parameters were generally found to be similar to previously published results by our groups and others. Samples from whole whiteflies and midguts of the B, A and Q biotypes that acquired TYLCV and SLCV were collected in both the US and Israel and hybridized to B. tabaci microarray. The data we analyzed, candidate genes that respond to both viruses in the three tested biotypes were identified and their expression that included quantitative real-time PCR and co-localization was verified for HSP70 by the Israeli group. In addition, experiments were undertaken to employ in situ hybridization to localize several candidate genes (in progress) using an oligonucleotide probe to the primary endosymbiont as a positive control. A proteome and corresponding transcriptome to enable more effective protein identification of adult whiteflies was constructed by the US group. Further validation of the transmission route of begomoviruses, mainly SLCV and the involvement of the digestive and salivary systems was investigated (Cicero and Brown). Due to time and budget constraints the RNAi-mediated silencing objective to verify gene function was not accomplished as anticipated. HSP70, a strong candidate protein that showed over-expression after TYLCV and SLCV acquisition and retention by B. tabaci, and co-localization with TYLCV in the midgut, was further studies. Besides this protein, our joint research resulted in the identification of many intriguing candidate genes and proteins that will be followed up by additional experiments during our future research. To identify these proteins it was necessary to increase the number and breadth of whitefly ESTs substantially and so whitefly cDNAs from various libraries made during the project were sequenced (Sanger, 454). As a result, the proteome annotation (ID) was far more successful than in the initial attempt to identify proteins using Uniprot or translated insect ESTs from public databases. The extent of homology shared by insects in different orders was surprisingly low, underscoring the imperative need for genome and transcriptome sequencing of homopteran insects. Having increased the number of EST from the original usable 5500 generated several years ago to >600,000 (this project+NCBI data mining), we have identified about one fifth of the whitefly proteome using these new resources. Also we have created a database that links all identified whitefly proteins to the PAVEdb-ESTs in the database, resulting in a useful dataset to which additional ESTS will be added. We are optimistic about the prospect of linking the proteome ID results to the transcriptome database to enable our own and other labs the opportunity to functionally annotate not only genes and proteins involved in our area of interest (whitefly mediated transmission) but for the plethora of other functionalities that will emerge from mining and functionally annotating other key genes and gene families in whitefly metabolism, development, among others. This joint grant has resulted in the identification of numerous candidate proteins involved in begomovirus transmission by B. tabaci. A next major step will be to capitalize on validated genes/proteins to develop approaches to interfere with the virus transmission.
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Jordan, Ramon L., Abed Gera, Hei-Ti Hsu, Andre Franck, and Gad Loebenstein. Detection and Diagnosis of Virus Diseases of Pelargonium. United States Department of Agriculture, July 1994. http://dx.doi.org/10.32747/1994.7568793.bard.

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Pelargonium (Geranium) is the number one pot plant in many areas of the United States and Europe. Israel and the U.S. send to Europe rooted cuttings, foundation stocks and finished plants to supply a certain share of the market. Geraniums are propagated mainly vegetatively from cuttings. Consequently, viral diseases have been and remain a major threat to the production and quality of the crop. Among the viruses isolated from naturally infected geraniums, 11 are not specific to Pelargonium and occur in other crops while 6 other viruses seem to be limited to geranium. However, several of these viruses are not sufficiently characterized to conclude that they are distinct agents and their nomenclature and taxonomy are confusing. The ability to separate, distinguish and detect the different viruses in geranium will overcome obstacles te developing effective detection and certification schemes. Our focus was to further characterize some of these viruses and develop better methods for their detection and control. These viruses include: isolates of pelargonium line pattern virus (PLPV), pelargonium ringspot virus (PelRSV), pelargonium flower break virus (PFBV), pelargonium leaf curl (PLCV), and tomato ringspot virus (TomRSV). Twelve hybridoma cell lines secreting monoclonal antibodies specific to a geranium isolate of TomRSV were produced. These antibodies are currently being characterized and will be tested for the ability to detect TomRSV in infected geraniums. The biological, biochemical and serological properties of four isometric viruses - PLPV, PelRSV, and PFBV (and a PelRSV-like isolate from Italy called GR57) isolated from geraniums exhibiting line and ring pattern or flower break symptoms - and an isolate ol elderbeny latent virus (ELV; which the literature indicates is the same as PelRSV) have been determined Cloned cDNA copies of the genomic RNAs of these viruses were sequenced and the sizes and locations of predicted viral proteins deduced. A portion of the putative replicase genes was also sequenced from cloned RT-PCR fragments. We have shown that, when compared to the published biochemical and serological properties, and sequences and genome organizations of other small isometric plant viruses, all of these viruses should each be considered new, distinct members of the Carmovirus group of the family Tombusviridae. Hybridization assays using recombinant DNA probes also demonstrated that PLPV, PelRSV, and ELV produce only one subgenomic RNA in infected plants. This unusual property of the gene expression of these three viruses suggests that they are unique among the Carmoviruses. The development of new technologies for the detection of these viruses in geranium was also demonstrated. Hybridization probes developed to PFBV (radioactively-labeled cRNA riboprobes) and to PLPV (non-radioactive digoxigenin-labeled cDNAs) were generally shown to be no more sensitive for the detection of virus in infected plants than the standard ELISA serology-based assays. However, a reverse transcriptase-polymerase chain reaction assay was shown to be over 1000 times more sensitive in detecting PFBV in leaf extracts of infected geranium than was ELISA. This research has lead to a better understanding of the identity of the viruses infecting pelargonium and to the development of new tools that can be used in an improved scheme of providing virus-indexed pelargonium plants. The sequence information, and the serological and cloned DNA probes generated from this work, will allow the application of these new tools for virus detection, which will be useful in domestic and international indexing programs which are essential for the production of virus-free germplasm both for domestic markets and the international exchange of plant material.
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