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Journal articles on the topic 'Curl-curl problem'

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Published: 9 March 2023

Last updated: 11 March 2023

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

Brenner, Susanne C., Jintao Cui, and Li-yeng Sung. "Multigrid Methods Based on Hodge Decomposition for a Quad-Curl Problem." Computational Methods in Applied Mathematics 19, no. 2 (April 1, 2019): 215–32. http://dx.doi.org/10.1515/cmam-2019-0011.

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AbstractIn this paper we investigate multigrid methods for a quad-curl problem on graded meshes. The approach is based on the Hodge decomposition. The solution for the quad-curl problem is approximated by solving standard second-order elliptic problems and optimal error estimates are obtained on graded meshes. We prove the uniform convergence of the multigrid algorithm for the resulting discrete problem. The performance of these methods is illustrated by numerical results.

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12

Brenner, Susanne C., Li-yeng Sung, and Jintao Cui. "An interior penalty method for a two dimensional curl-curl and grad-div problem." ANZIAM Journal 50 (July 1, 2009): 947. http://dx.doi.org/10.21914/anziamj.v50i0.1600.

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13

Chen, Jun, and Xing-Bin Pan. "Quasilinear systems involving curl." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 2 (December 28, 2017): 243–79. http://dx.doi.org/10.1017/s0308210517000014.

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We study a quasilinear degenerately elliptic system involving the operator curl. The leading-order term of the associated energy functional is a q-power of curl of the unknowns. It is interesting to us that the structure of the lower-order terms will play an important role in both the existence and regularity of the solutions. When the lower-order part of the energy functional is convex we obtain weak solutions by minimizing the functional in some suitable spaces of vector fields. When it is concave we obtain critical points of the truncated functional, which are weak solutions of a nonlinear eigenvalue problem. We also examine the interior Hölder regularity of the weak solutions, which indicates that the weak solution is composed of a ‘good’ divergence-free part and a ‘bad’ part in gradient form. The analysis involves local higher integrability and local Hölder gradient estimates for a translated q′-Laplace equation and a translated quasilinear equation with p-growth.

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14

Brenner, S. C., J. Cui, F. Li, and L. Y. Sung. "A nonconforming finite element method for a two-dimensional curl–curl and grad-div problem." Numerische Mathematik 109, no. 4 (April 26, 2008): 509–33. http://dx.doi.org/10.1007/s00211-008-0149-7.

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15

BENBOURHIM, M. N., and A. BOUHAMIDI. "DIV-CURL WEIGHTED MINIMIZING SPLINES." Analysis and Applications 05, no. 02 (April 2007): 95–122. http://dx.doi.org/10.1142/s0219530507000900.

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The paper deals with a div-curl approximation problem by weighted minimizing splines. The weighted minimizing splines are an extension of the well-known thin plate splines and are radial basis functions which allow the approximation or the interpolation of a scalar function from given scattered data. In this paper, we show that the theory of the weighted minimizing splines may also be used for the approximation or for the interpolation of a vector field controlled by the divergence and the curl of the vector field. Numerical examples are given to show the efficiency of this method.

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16

Zhang, Qian, Lixiu Wang, and Zhimin Zhang. "H(curl$^2$)-Conforming Finite Elements in 2 Dimensions and Applications to the Quad-Curl Problem." SIAM Journal on Scientific Computing 41, no. 3 (January 2019): A1527—A1547. http://dx.doi.org/10.1137/18m1199988.

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17

Hardiyanty, Nur, Ajeng Kartini Mas'ud, and Ita Rini. "PENGARUH PEMBERIAN TOWEL CURL EXERCISE TERHADAP KELINCAHAN ANAK FLAT FOOT USIA 7-9 TAHUN DI MAKASSAR EFFECT OF TOWEL CURL EXERCISE ON THE AGILITY OF FLAT CHILDREN FOOT AGES 7-9 YEARS IN MAKASSAR." Jurnal Fisioterapi dan Rehabilitasi 4, no. 2 (August 12, 2020): 67–73. http://dx.doi.org/10.33660/jfrwhs.v4i2.117.

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ABSTRAK Anak flat foot usia 7-9 tahun berisiko memiliki gangguan kelincahan dibandingkan anak normal usia 7-9 tahun lainnya. Penelitian ini mengangkat permasalahan pengaruh pemberian towel curl exercise terhadap kelincahan anak flat foot usia 7-9 tahun di makassar. Tujuan penelitian ini adalah untuk mengetahui permasalahan pengaruh pemberian towel curl exercise terhadap kelincahan anak flat foot usia 7-9 tahun di makassar. Jenis penelitian yang digunakan adalah pre-experimental dengan menggunakan desain penelitian one-group pretest posttest design dengan variabel independent adalah towel curl exercise dan variabel dependent adalah kelincahan. Teknik pengambilan sampel menggunakan teknik purposive sampling dengan jumlah sampel 29 orang. Penentuan sampel berdasarkan kriteria inklusi dan eksklusi. Instrumen atau alat pengambilan data yang digunakan dalam penelitian ini adalah shuttle run test . Penelitian ini dilakukan selama 5 minggu. Hasil penelitian menunjukkan bahwa 29 responden yang diberikan towel curl exercise dengan frekuensi 3 kali dalam 1 minggu berpengaruh terhadap perubahan kelincahan anak flat foot usia 7-9 tahun di makassar dengan nilai signifikan Uji T berpasangan P = 0.000 (p<0,05). Penelitian ini menunjukkan terdapat pengaruh towel curl exercise terhadap kelincahan anak flat foot usia 7-9 tahun di makassar. Kata Kunci : Towel Curl Exercise, Kelincahan, Flat Foot ABSTRACT Flat foot children aged 7-9 years are at risk of having agility disorders compared to normal children aged 7-9 years. This study raised the issue of the effect of giving curl towel exercise to the agility of flat foot children aged 7-9 years in Makassar. The purpose of this study was to determine the problem of the effect of giving curl towel exercise to the agility of flat foot children aged 7-9 years in Makassar. The type of research used was pre-experimental using a one-group pretest posttest design design with independent variables being towel curl exercise and the dependent variable was agility. The sampling technique used purposive sampling technique with a sample of 29 people. Determination of samples based on inclusion and exclusion criteria. The instrument or data collection tool used in this research is shuttle run test. This study was conducted for 5 weeks. The results showed that 29 respondents who were given curl towel exercise with a frequency of 3 times a week had an effect on changes in agility of flat foot children aged 7-9 years in Makassar with a significant value of paired T Test P = 0.000 (p <0.05). This study showed that there was an effect of towel curl exercise on the agility of flat foot children aged 7-9 years in Makassar. Keywords: Towel Curl Exercise, Agility, Flat Foot

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18

Hussain, Muhammad S., Khalid Naveed, and Muhammad Atiq. "SUSCEPTIBILITY OF CHILLI LINES / VARIETIES TOWARDS CHILLI LEAF CURL VIRUS AND ITS MANAGEMENT THROUGH VECTOR CONTROL." Pakistan Journal of Phytopathology 29, no. 1 (July 12, 2017): 17. http://dx.doi.org/10.33866/phytopathol.029.01.0304.

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Chilli leaf curl virus (ChiLCV) is a serious problem for chilli production in Pakistan and India. In this study, fourteen different chilli lines / varieties were screened for their resistance/susceptibility towards ChiLCV in open field trials. Data of disease severity and disease incidence was recorded on weekly basis. Out of all varieties screened, Tatapuri Chilli and CH111 showed high susceptibility towards the chilli leaf curl disease. Talhari, CH 99, CH103, CH 106, CH107, CH108, CH109, GSL111 showed susceptible response whereas CBS1292 showed moderately susceptible response towards the disease. Only two cultivars Hybrid-46 and Hot Queen were found as moderately resistant. The screening experiment showed that most of the chilli varieties are susceptible to chilli leaf curl disease and resistance is lacking in local germplasm. Furthermore, three insecticides: Imidacloprid, Polo and Emamectin were tested for their efficacy for the management of disease and whitefly vector in field conditions at weekly intervals. Out of three insecticides, Polo gave best results as compared to Emamectin and Imidacloprid against chilli leaf curl disease and whitefly vector.

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19

Moroşanu, Georgiana-Alexandra, Virgil-Gabriel Teodor, and Nicolae Oancea. "Profiling of the Generating Rack of an Ordered Curl Known in Discrete Form by the Virtual Pole Method." Bulletin of the Polytechnic Institute of Iași. Machine constructions Section 67, no. 4 (December 1, 2021): 25–34. http://dx.doi.org/10.2478/bipcm-2021-0020.

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Abstract Situations encountered in “reverse engineering” are relatively common in which, to address a problem of profiling generating tools for processing an orderly curl of surfaces (cylindrical or helical teeth), data known by measuring the existing product are used. The surface numerical representation can be done in the form of a points cloud. The profiling algorithm proposed in the paper is based on the possibility of measuring surfaces and their representation through an ordered cloud of points. The algorithm uses the complementary theorem of the virtual pole, for profiling the rack tool generating a piece that has an ordered curl of profiles. The methodology is not limited to the profiling of the rack tool but can be extended to the problem of profiling the primary peripheral surface of some screw-type tools, generating the ordered curl of surfaces. The paper also presents a numerical application developed based on a specific software product.

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20

Chung, Nguyen Thanh. "Existence of solutions for a class of $p(x)$-curl systems arising in electromagnetism without (A-R) type conditions." Tamkang Journal of Mathematics 51, no. 3 (September 25, 2020): 187–200. http://dx.doi.org/10.5556/j.tkjm.51.2020.2915.

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In this paper, we study the existence and multiplicity of solutions for a class of of $p(x)$-curl systems arising in electromagnetism. Under suitable conditions on the nonlinearities which do not satisfy Ambrosetti-Rabinowitz type conditions, we obtain some existence and multiplicity results for the problem by using the mountain pass theorem and fountain theorem. Our main results in this paper complement and extend some earlier ones concerning the $p(x)$-curl operator in [4, 15].

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21

Wan, Andy T. S., and Marc Laforest. "A Posteriori Error Estimation for the p-Curl Problem." SIAM Journal on Numerical Analysis 58, no. 1 (January 2020): 460–91. http://dx.doi.org/10.1137/16m1075624.

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22

Sun, Jiguang. "A mixed FEM for the quad-curl eigenvalue problem." Numerische Mathematik 132, no. 1 (February 20, 2015): 185–200. http://dx.doi.org/10.1007/s00211-015-0708-7.

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23

Liu, Jie, Xuanfeng Liu, Yongxin Jiang, Xin Zhou, Li Zhang, and Xuenong Wang. "Research on the Adaptability of High-Performance Film for Full Recycling to the Curl-Up Film Collecting Method." Agriculture 12, no. 7 (July 19, 2022): 1051. http://dx.doi.org/10.3390/agriculture12071051.

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Given the problem of the low tensile performance of the plastic film used in China, which brings about difficulties in curl-up film collecting, in this study, a contrast test was carried out on the tensile property of high-performance film for full recycling and the ordinary polyethylene film (PE film) that is used extensively in China. Test results showed that, within the service period, the elongation at break and tensile yield stress of the high-performance film were higher than those of ordinary polyethylene film, and, within the film-laying period of 0~30 days, the reduction scale of the elongation at break and tensile yield stress was higher than that within the film-laying period of 30~180 days. In this study, in order to obtain the lowest tensile performance of the film by curl-up film collecting, the operation principles of the curl-up film collectors were analyzed. The test on the force of curling up the film in the process of overcoming the force between the film and soil was analyzed. Test and analysis results showed that, for different sampling positions, film pick-up angles, and film types, the tensile stress on the film while pulling it up was within a range of 15.97~21.86 MPa. In order to verify the curling up effect of differently structured film collectors on different types of film with different thicknesses, a field test on film curl-up collecting was designed. A contrast test was carried out on two types of curl-up film collectors, 1JRM-2000 and 11SM-1.2, and the test results showed that the film recycling rate and working performance on the film laid in the same year by the film collector with a fixed film pick-up angle were higher than those for varying film pick-up angles. The curl-up film collector fixed with an automatic film-guiding mechanism is not affected by the velocity difference between the linear velocity of the film curl-up mechanism and the advancing velocity of the machine. The film recycling rate and working performance on the film laid in the same year by the 11SM-1.2 curl-up film collector can meet the operational requirements for collecting high-performance film with thicknesses of 0.008 mm and 0.01 mm. This research can provide a reference for simplifying the structure of residual plastic film collectors, increasing the film recycling rate, and reducing the cost.

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24

Lieberman, Gary, and Xing-Bin Pan. "On a quasilinear system arising in the theory of superconductivity." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 141, no. 2 (April 2011): 397–407. http://dx.doi.org/10.1017/s0308210509001395.

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We examine the regularity of the solution of a quasilinear system involving the curl of vector fields. This system arises in the mathematical theory of superconductivity. The C2+α regularity was obtained by Bates and Pan under the condition that Ω is simply connected and has no holes, and that the normal component of the curl of the boundary data vanishes. The aim of this paper is to remove these technical restrictions on the topology of the domain and on the boundary data. By carefully studying the related quasilinear Neumann problem, we obtain the C2+α regularity without assuming these technical conditions.

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25

Wang, Lixiu, Weikun Shan, Huiyuan Li, and Zhimin Zhang. "H(curl2)-conforming quadrilateral spectral element method for quad-curl problems." Mathematical Models and Methods in Applied Sciences 31, no. 10 (September 2021): 1951–86. http://dx.doi.org/10.1142/s0218202521500433.

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In this paper, we propose an [Formula: see text]-conforming quadrilateral spectral element method to solve quad-curl problems. Starting with generalized Jacobi polynomials, we first introduce quasi-orthogonal polynomial systems for vector fields over rectangles. [Formula: see text]-conforming elements over arbitrary convex quadrilaterals are then constructed explicitly in a hierarchical pattern using the contravariant transform together with the bilinear mapping from the reference square onto each quadrilateral. It is worth noting that both the simplest rectangular and quadrilateral spectral elements possess only 8 degrees of freedom on each physical element. In the sequel, we propose our [Formula: see text]-conforming quadrilateral spectral element approximation based on the mixed weak formulation to solve the quad-curl equation and its eigenvalue problem. Numerical results show the effectiveness and efficiency of our method.

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26

Lamberti, Pier Domenico, and Michele Zaccaron. "Spectral stability of the curlcurl operator via uniform Gaffney inequalities on perturbed electromagnetic cavities." Mathematics in Engineering 5, no. 1 (2022): 1–31. http://dx.doi.org/10.3934/mine.2023018.

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<abstract><p>We prove spectral stability results for the $ curl curl $ operator subject to electric boundary conditions on a cavity upon boundary perturbations. The cavities are assumed to be sufficiently smooth but we impose weak restrictions on the strength of the perturbations. The methods are of variational type and are based on two main ingredients: the construction of suitable Piola-type transformations between domains and the proof of uniform Gaffney inequalities obtained by means of uniform a priori $ H^2 $-estimates for the Poisson problem of the Dirichlet Laplacian. The uniform a priori estimates are proved by using the results of V. Maz'ya and T. Shaposhnikova based on Sobolev multipliers. Connections to boundary homogenization problems are also indicated.</p></abstract>

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27

Hong, Qingguo. "A Discontinuous Galerkin Method for the Fourth-Order Curl Problem." Journal of Computational Mathematics 30, no. 6 (June 2012): 565–78. http://dx.doi.org/10.4208/jcm.1206-m3572.

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28

Bertagnolio, F., and O. Daube. "Solution of the Div-Curl Problem in Generalized Curvilinear Coordinates." Journal of Computational Physics 138, no. 1 (November 1997): 121–52. http://dx.doi.org/10.1006/jcph.1997.5800.

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29

Wakamatsu, Tsuyoshi, Michael G. G. Foreman, Patrick F. Cummins, and Josef Y. Cherniawsky. "On the Influence of Random Wind Stress Errors on the Four-Dimensional, Midlatitude Ocean Inverse Problem." Monthly Weather Review 137, no. 6 (June 1, 2009): 1844–62. http://dx.doi.org/10.1175/2008mwr2621.1.

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Abstract The effects of the parameterized wind stress error covariance function on the a priori error covariance of an ocean general circulation model (OGCM) are examined. These effects are diagnosed by computing the projection of the a priori model state error covariance matrix to sea surface height (SSH). The sensitivities of the a priori error covariance to the wind stress curl error are inferred from the a priori SSH error covariance and are shown to differ between the subpolar and subtropical gyres because of different contributions from barotropic and baroclinic ocean dynamics. The spatial structure of the SSH error covariance due to the wind stress error indicates that the a priori model state error is determined indirectly by the wind stress curl error. The impact of this sensitivity on the solution of a four-dimensional inverse problem is inferred.

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30

Delgado, Berenice B., and Jorge Eduardo Macías-Díaz. "An Exterior Neumann Boundary-Value Problem for the Div-Curl System and Applications." Mathematics 9, no. 14 (July 8, 2021): 1609. http://dx.doi.org/10.3390/math9141609.

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We investigate a generalization of the equation curlw→=g→ to an arbitrary number n of dimensions, which is based on the well-known Moisil–Teodorescu differential operator. Explicit solutions are derived for a particular problem in bounded domains of Rn using classical operators from Clifford analysis. In the physically significant case n=3, two explicit solutions to the div-curl system in exterior domains of R3 are obtained following different constructions of hyper-conjugate harmonic pairs. One of the constructions hinges on the use of a radial integral operator introduced recently in the literature. An exterior Neumann boundary-value problem is considered for the div-curl system. That system is conveniently reduced to a Neumann boundary-value problem for the Laplace equation in exterior domains. Some results on its uniqueness and regularity are derived. Finally, some applications to the construction of solutions of the inhomogeneous Lamé–Navier equation in bounded and unbounded domains are discussed.

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Elbaz, Mounira, Monaam Timoumi, and Peter Hanson. "Behavior of new entries and developed tomato hybrids carrying Ty-2 genes." Tunisian Journal of Plant Protection 17, no. 1 (June 30, 2022): 1–14. http://dx.doi.org/10.52543/tjpp.17.1.1.

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Tomato yellow leaf curl disease (TYLCD) is a serious problem hampering tomato production worldwide. In the Mediterranean Basin, disease incidence and severity are higher in the dry season increasing whitefly (Bemisia tabaci) populations. Effectiveness of resistance to Tomato yellow leaf curl virus (TYLCV) depends on both tomato host resistance and TYLCV complex species. So far, six different Ty tomato resistance genes have been identified. Two main TYLCV complex species, Tomato yellow leaf curl virus-Israel (TYLCV-Is) and Tomato yellow leaf curl Sardinia virus (TYLCSV), have been identified in Tunisia. The present work aimed to evaluate entries heterozygous for Ty-2 gene to help predict hybrid performance. Two tomato entries homozygous for the Ty-2 TYLCV resistance gene, one tomato hybrid homozygous for Ty-2 and two heterozygous hybrids were included, besides two susceptible tomato entries. Resistance response to TYLCD was recorded based on disease incidence and severity levels. Data analysis was performed according to presence/absence of Ty-2 gene and taking into account homozygosity and heterozygosity of Ty-2. Generalized linear model analysis was applied to check significance of individual factors' effects (i.e. effect of tomato entries or tomato groups of entries based on presence or absence of homozygous/heterozygous Ty-2 gene, block unit within the field trial and the year of the trial) on the dependent variables (disease incidence and severity). Further multicomparison tests gave evidence on significant effect of Ty-2 homozygous gene tomato entries on TYLCD incidence and severity levels. The results were discussed with special focus on the relevance use of heterozygous hybrid tomato varieties.

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Liu, Li, and Jyhwen Wang. "Modeling Springback of Metal-Polymer-Metal Laminates." Journal of Manufacturing Science and Engineering 126, no. 3 (August 1, 2004): 599–604. http://dx.doi.org/10.1115/1.1765141.

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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 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 wiper die bending. Based on the integration of a straight beam and a curved beam models, the springback factor Ks is calculated. It is shown that the prediction is in good agreement with the published experimental data. Application of the integrated model to minimize the springback and side wall curl is demonstrated. The analytical model leads to a simple expression that predicts the springback factor. The ability to predict the shape analytically is significant, since other methods require extensive finite element simulation of the deformation process.

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Sun, Zhengjia, Fuzheng Gao, Chao Wang, and Yi Zhang. "A QUADRATIC C0 INTERIOR PENALTY METHOD FOR THE QUAD-CURL PROBLEM." Mathematical Modelling and Analysis 25, no. 2 (March 18, 2020): 208–25. http://dx.doi.org/10.3846/mma.2020.9796.

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In this paper we study the C0 interior penalty method for a quad-curl problem arising from magnetohydrodynamics model on bounded polygons or polyhedrons. We prove the well-posedness of the numerical scheme and then derive the optimal error estimates in a discrete energy norm. A post-processing procedure that can produce C1 approximations is also presented. The performance of the method is illustrated by numerical experiments.

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Glick, E., Y. Levy, and Y. Gafni. "The viral etiology of tomato yellow leaf curl disease – a review." Plant Protection Science 45, No. 3 (October 16, 2009): 81–97. http://dx.doi.org/10.17221/26/2009-pps.

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Tomato yellow leaf curl disease (TYLCD) is one of the most devastating plant diseases in the world. As a result of its continuing rapid spread, it now afflicts more than 30 tomato growing countries in the Mediterranean basin, southern Asia, Africa, and South, Central and North America. The disease is caused by a group of viral species of the genus <I>Begomovirus,</I> family Geminiviridae (geminiviruses), referred to as <I>Tomato yellow leaf curl virus</I> (TYLCV). These are transmitted by an insect vector, the whitefly<I> Bemisia tabaci</I>, classified in the family Aleyrodidae. The genome of TYLCV generally consists of a single circular single-stranded (ss) DNA molecule, with only one exception in which two components were identified. It encodes six open reading frames, only one of which codes for the coat protein (CP) that represents a building block of the viral particle. TYLCV, like all other members of the Geminiviridae, has geminate particles, apparently consisting of two incomplete T = 1 icosahedra joined together to produce a structure with 22 pentameric capsomers and 110 identical CP subunits. Close to 50 years of intensive research into TYLCV epidemics has been conducted to find solutions to the severe problem caused by this virus. To date, breeding for resistance appears to be the best approach to controlling this disease, although only partially resistant varieties are commercially available. Since the virus consists of a ssDNA that replicates in the host-cell nucleus, the molecular mechanisms involved in its nuclear import have been the focus of our studies in recent years and results, as well as prospects, are discussed in this review. In addition, we describe our recent finding of a suppressor of gene silencing encoded by one of the TYLCV-Isr genes. This paper provides an overview of the most outstanding achievements in TYLCV research that may lead to more effective control strategies.

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Brenner, Susanne C., Jiguang Sun, and Li-yeng Sung. "Hodge Decomposition Methods for a Quad-Curl Problem on Planar Domains." Journal of Scientific Computing 73, no. 2-3 (May 20, 2017): 495–513. http://dx.doi.org/10.1007/s10915-017-0449-0.

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36

Bensow, Rickard, and Mats G. Larson. "Discontinuous Least-Squares finite element method for the Div-Curl problem." Numerische Mathematik 101, no. 4 (September 15, 2005): 601–17. http://dx.doi.org/10.1007/s00211-005-0600-y.

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37

Muijs, Remco, Klaus Holliger, and Johan O. A. Robertsson. "Perturbation analysis of an explicit wavefield separation scheme for P‐ and S‐waves." GEOPHYSICS 67, no. 6 (November 2002): 1972–82. http://dx.doi.org/10.1190/1.1527097.

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Dense spatial recording patterns of three‐component (3C) receivers allow for direct wavefield decomposition through explicit calculation of divergence and curl of the recorded elastic wavefield. Since this approach is based upon the observation of small phase shifts, it requires highly accurate deployment of the receiver configurations. To study the feasibility of a recently proposed P/S‐wave separation scheme, we systematically assess the effects of position and orientation errors of one or several geophones within the recording pattern on technique performance. We find that realistic deployment errors can significantly affect estimates of the divergence and curl of particle velocity. The errors induced by mispositioned or misoriented geophones differ for each of the geophones that make up a pattern. Moreover, the inaccuracies vary with the angle of incidence, potentially affecting analysis procedures applied to the data at a later stage, such as amplitude variation with offset (AVO). Based on a relative L1‐criterion, the position of each receiver needs to be accurate within 10% of the length of the sides of the configuration to obtain meaningful divergence and curl estimates. Furthermore, the output is particularly sensitive to misorientations of geophones, requiring that the orientations of all geophones be accurate within 2°. These observations point to significant difficulties when applying this technique. To alleviate this problem, we present an approach to detect and compensate for such deployment‐related inaccuracies prior to explicit P/S‐wave separation. This strategy is based on a pyramid‐shaped receiver configuration and relies on minimizing the differences between the divergence and curl estimates calculated over the pyramid and each of the four subtetrahedra that comprise the pyramid.

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38

Aramaki, Junichi. "Minimizing the Lp norm of the curl of vector fields in a multi-connected domain." International Journal of Mathematics 28, no. 01 (January 2017): 1750004. http://dx.doi.org/10.1142/s0129167x17500045.

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We study the problem of minimizing the [Formula: see text] norm of the curl of vector fields in a three-dimensional, bounded multi-connected domain with a prescribed tangential component on the boundary. We then prove the existence of minimizers and estimate them.

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39

Alexandre Journeaux, Antoine, Nicolas Nemitz, and Olivier Moreau. "Locally conservative projection methods: benchmarking and practical implementation." COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 33, no. 1/2 (December 20, 2013): 663–87. http://dx.doi.org/10.1108/compel-03-2013-0091.

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Purpose – This article presents a locally conservative projection method which aims to preserve the integral of a function and one operator among grad, div, or curl. Design/methodology/approach – After a theoretical description of the projection methods, the locally conservative projection is analytically tested and compared with the orthogonal method. In the second part, the implementation of the methods is described, and improvements are proposed. An industrial application of the present work, consisting in a magneto-thermal coupled problem, is then presented. Findings – The implementation of the conservative method is simpler than the implementation of the orthogonal method while presenting similar behaviour in terms of accuracy and conservation. Originality/value – The locally conservative method is extended to curl-conform and div-conform elements. Furthermore, three-dimensional studies are proposed.

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40

ZaharaddeenSamaila, Gilima, David Kashina Boniface, Olalekan Oyeleke Banwo, Alegbejo Mathew Dada, Charles Chindo Agart, and Ibrahim Tijjani. "Occurrence and Distribution of Viruses Associated with Okra and Their Alternative Hosts in Kaduna and Zamfara States, Nigeria." Journal of Tropical Crop Science 8, no. 03 (October 20, 2021): 177–86. http://dx.doi.org/10.29244/jtcs.8.03.177-186.

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One of the major constraints to production of okra (Abelmoschus esculentus L.) in Nigeria and in particular in Kaduna and Zamfara States, is the problem of okra mosaic virus and okra leaf curl virus. This study was carried out to provide information on the occurrence and distribution of okra mosaic and okra leaf curl viruses on okra, in Kaduna and Zamfara states, Nigeria. A survey of okra-producing farms was carried out during dry and wet seasons of 2017 cropping season in Kaduna (Zaria, Lere, and Igabi Local Government Areas) and Zamfara (Gusau, Bungudu, and Zurmi LGAs) states. Leaf samples (15) of symptomatic okra plants were collected from each farm in the study area. The total number of plants and the number of symptomatic plants within each subplot were recorded, and the disease incidence was determined. Double Antibody Sandwich Enzyme-Linked Immunosorbent Assay (DAS- ELISA) kit was used in the detection of Okra Mosaic Virus while Polymerase Chain Reaction (PCR) was employed for the detection Okra Leaf Curl Virus. The results showed that all the okra leaf samples tested for OLCV were negative in this study while OkMV was tested positve in all the samples with a recorded incidence of 20 % and 14 % in Kaduna and Zamfara states respectively, however, only 8 out of total weed samples were also tested positive for OKV, but all were tested negative to OLCV.

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41

NEFF, PATRIZIO, KRZYSZTOF CHEŁMIŃSKI, and HANS-DIETER ALBER. "NOTES ON STRAIN GRADIENT PLASTICITY: FINITE STRAIN COVARIANT MODELLING AND GLOBAL EXISTENCE IN THE INFINITESIMAL RATE-INDEPENDENT CASE." Mathematical Models and Methods in Applied Sciences 19, no. 02 (February 2009): 307–46. http://dx.doi.org/10.1142/s0218202509003449.

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We propose a model of finite strain gradient plasticity including phenomenological Prager type linear kinematical hardening and nonlocal kinematical hardening due to dislocation interaction. Based on the multiplicative decomposition, a thermodynamically admissible flow rule for Fp is described involving as plastic gradient Curl Fp. The formulation is covariant w.r.t. superposed rigid rotations of the reference, intermediate and spatial configuration but the model is not spin-free due to the nonlocal dislocation interaction and cannot be reduced to a dependence on the plastic metric [Formula: see text]. The linearization leads to a thermodynamically admissible model of infinitesimal plasticity involving only the Curl of the nonsymmetric plastic distortion p. Linearized spatial and material covariance under constant infinitesimal rotations is satisfied. Uniqueness of strong solutions of the infinitesimal model is obtained if two non-classical boundary conditions on the plastic distortion p are introduced: [Formula: see text] on the microscopically hard boundary ΓD ⊂ ∂Ω and [ Curl p] · τ = 0 on the microscopically free boundary ∂Ω\ΓD, where τ are the tangential vectors at the boundary ∂Ω. A weak reformulation of the infinitesimal model allows for a global in-time solution of the rate-independent initial boundary value problem. The method is based on a mixed variational inequality with symmetric and coercive bilinear form. We use a new Hilbert-space suitable for dislocation density dependent plasticity.

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Cai, Zhiqiang, Shuhao Cao, and Rob Falgout. "Robust a posteriori error estimation for finite element approximation to (curl) problem." Computer Methods in Applied Mechanics and Engineering 309 (September 2016): 182–201. http://dx.doi.org/10.1016/j.cma.2016.06.007.

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43

Han, Jiayu. "Shifted inverse iteration based multigrid methods for the quad-curl eigenvalue problem." Applied Mathematics and Computation 367 (February 2020): 124770. http://dx.doi.org/10.1016/j.amc.2019.124770.

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44

Bolinder, C. J. "Curvilinear Coordinates and Physical Components: An Application to the Problem of Viscous Flow and Heat Transfer in Smoothly Curved Ducts." Journal of Applied Mechanics 63, no. 4 (December 1, 1996): 985–89. http://dx.doi.org/10.1115/1.2787256.

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Expressions are derived for the gradient, divergence, Laplacian, curl, and material time derivative in terms of general curvilinear coordinates using physical components of all vector quantities. The results are conveniently expressed in terms of two new coefficients, involving physical and reciprocal base vectors. An application to the problem of viscous flow and heat transfer in arbitrarily smoothly curved ducts is presented. In particular, helical ducts are considered.

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de Felice, G., F. S. Marra, and G. C. Rufolo. "Regularized solutions for the discrete forms of the div–curl problem in CFD." Computing and Visualization in Science 4, no. 3 (February 2002): 175–82. http://dx.doi.org/10.1007/s007910100069.

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Sun, Zhengjia, Jintao Cui, Fuzheng Gao, and Chao Wang. "Multigrid methods for a quad-curl problem based on C0 interior penalty method." Computers & Mathematics with Applications 76, no. 9 (November 2018): 2192–211. http://dx.doi.org/10.1016/j.camwa.2018.07.048.

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47

Chi, Jieru, Lei Guo, Aurelien Destruel, Yaohui Wang, Chunyi Liu, Mingyan Li, Ewald Weber, et al. "Magnetic Resonance-Electrical Properties Tomography by Directly Solving Maxwell’s Curl Equations." Applied Sciences 10, no. 9 (May 10, 2020): 3318. http://dx.doi.org/10.3390/app10093318.

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Magnetic Resonance-Electrical Properties Tomography (MR-EPT) is a method to reconstruct the electrical properties (EPs) of bio-tissues from the measured radiofrequency (RF) field in Magnetic Resonance Imaging (MRI). Current MR-EPT approaches reconstruct the EP profile by solving a second-order partial differential wave equation problem, which is sensitive to noise and can induce large reconstruction artefacts near tissue boundaries and areas with inaccurate field measurements. In this paper, a novel MR-EPT approach is proposed, which is based on a direct solution to Maxwell’s curl equations. The distribution of EPs is calculated by iteratively fitting the RF field calculated by the finite-difference-time-domain (FDTD) technique to the measured values. To solve the time-consuming problem of the iterative fitting, a graphics processing unit (GPU) is used to accelerate the FDTD technique to process the field calculation kernel. The new EPT method was evaluated by investigating a numerical head phantom, and it was found that EP values can be accurately calculated and were relatively insensitive to simulated RF field errors. The feasibility of the proposed method was further validated using phantom-based experimental data collected from a 9.4 Tesla (T) Magnetic Resonance Imaging (MRI) system. The results also indicated that more accurate EP values could be reconstructed from the new method compared with conventional methods. Moreover, even in the absence of phase information of the RF field, the proposed approach is still capable of offering robust EPT solutions, thus having improved practicality for potential clinical applications.

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48

ZHU, JIANG, JEAN-LUC GUERMOND, ABIMAEL F. D. LOULA, and LUIGI QUARTAPELLE. "3D VECTOR POISSON-LIKE PROBLEM WITH A TRIPLET OF INTRINSIC SCALAR BOUNDARY CONDITIONS." Mathematical Models and Methods in Applied Sciences 13, no. 12 (December 2003): 1725–43. http://dx.doi.org/10.1142/s0218202503003094.

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In this work, we consider the three-dimensional vector Poisson-like equation supplemented by a nonstandard set of three scalar boundary conditions consisting of the simultaneous specification of the divergence of the unknown, its normal component, and the normal component of its curl on the entire boundary. A weak formulation of this elliptic boundary value problem is proposed. Existence and uniqueness of a solution are established under two compatibility conditions. An uncoupled solution algorithm is introduced together with its finite element approximation. The corresponding error analysis is performed.

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Ren, Zhengyong, Thomas Kalscheuer, Stewart Greenhalgh, and Hansruedi Maurer. "A finite-element-based domain-decomposition approach for plane wave 3D electromagnetic modeling." GEOPHYSICS 79, no. 6 (November 1, 2014): E255—E268. http://dx.doi.org/10.1190/geo2013-0376.1.

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We developed a novel parallel domain-decomposition approach for 3D large-scale electromagnetic induction modeling in the earth. We used the edge-based finite-element method and unstructured meshes. Unstructured meshes were divided into sets of nonoverlapping subdomains. We used the curl-curl electric field equation to carry out the analysis. In each subdomain, the electric field was discretized by first-order vector shape functions along the edges of tetrahedral elements. The tangential components of the magnetic field on the interfaces of the subdomains were defined as a set of Lagrange multipliers. The unknown Lagrange multipliers were solved from an interface problem defined on the interfaces of the subdomains. With the availability of the Lagrange multipliers, the electric field values in each subdomain were solved independently. Three synthetic examples were evaluated to verify our code. Excellent agreement with previously published solutions was obtained. Synthetic examples revealed that our domain decomposition technique is scalable with respect to the number of subdomains and robust with regard to frequency and the heterogeneous distribution of material parameters, i.e., electric conductivity, electric permittivity, and magnetic permeability.

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Mora, Jaime, and Leszek Demkowicz. "Fast Integration of DPG Matrices Based on Sum Factorization for all the Energy Spaces." Computational Methods in Applied Mathematics 19, no. 3 (July 1, 2019): 523–55. http://dx.doi.org/10.1515/cmam-2018-0205.

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AbstractNumerical integration of the stiffness matrix in higher-order finite element (FE) methods is recognized as one of the heaviest computational tasks in an FE solver. The problem becomes even more relevant when computing the Gram matrix in the algorithm of the Discontinuous Petrov Galerkin (DPG) FE methodology. Making use of 3D tensor-product shape functions, and the concept of sum factorization, known from standard high-order FE and spectral methods, here we take advantage of this idea for the entire exact sequence of FE spaces defined on the hexahedron. The key piece to the presented algorithms is the exact sequence for the one-dimensional element, and use of hierarchical shape functions. Consistent with existing results, the presented algorithms for the integration of {H^{1}}, {H(\operatorname{curl})}, {H(\operatorname{div})}, and {L^{2}} inner products, have the {\mathcal{O}(p^{7})} computational complexity in contrast to the {\mathcal{O}(p^{9})} cost of conventional integration routines. Use of Legendre polynomials for shape functions is critical in this implementation. Three boundary value problems under different variational formulations, requiring combinations of {H^{1}}, {H(\operatorname{div})} and {H(\operatorname{curl})} test shape functions, were chosen to experimentally assess the computation time for constructing DPG element matrices, showing good correspondence with the expected rates.

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