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Electro-Geometrical Sensitivity Analysis of Electromagnetic Cavity BP-NGD Equalization 电磁腔 BP-NGD 均衡的电几何灵敏度分析
IF 2.3 Q1 Mathematics Pub Date : 2024-02-20 DOI: 10.1109/JMMCT.2024.3367604
Blaise Ravelo;Hongyu Du;Glauco Fontgalland;Fayu Wan
This article considers an electro-thermo-geometrical Multiphysics analysis of electromagnetic compatibility (EMC) resonance problem solution by using bandpass (BP) type negative group delay (NGD) equalization method. The rectangular cavity electric model based on EMC frequency domain S-parameter analysis is introduced. The unfamiliar BP-NGD function is specified in order to size the lumped electrical components of the suitable RLC-network based topology. The BP-NGD equalization principle is described including the Multiphysics synoptic analysis by means of electro- thermo-geometrical approach of the problem. The BP-NGD equalization methodology is proposed. The feasibility study of the EMC resonance equalization method is validated by considering a proof-of-concept constituted by 232.9×28×3.8 cm-size rectangular cavity. The BP-NGD active circuit is designed as equalizer by using RLC-series network. The EMC solution is verified by the BP-NGD POC specified by −4 ns NGD value at 0.644 MHz center frequency stating resonance effect reduction with 1-dB flatness. Furthermore, time-domain signal integrity (SI) analysis confirms the EMC cavity resonance resolution by showing output delay, over/under shoot reduction and also input-output cross correlation improvement from 89% to 99%.
本文探讨了利用带通(BP)型负群延迟(NGD)均衡方法解决电磁兼容(EMC)共振问题的电热几何多物理场分析。介绍了基于 EMC 频域 S 参数分析的矩形腔电气模型。为了确定合适的基于 RLC 网络拓扑结构的块状电气元件的尺寸,指定了陌生的 BP-NGD 函数。介绍了 BP-NGD 均衡原理,包括通过电热几何方法对问题进行多物理场同步分析。提出了 BP-NGD 均衡方法。通过对 232.9×28×3.8 厘米大小的矩形腔进行概念验证,验证了电磁兼容共振均衡方法的可行性研究。使用 RLC 串联网络设计了 BP-NGD 有源电路作为均衡器。在 0.644 MHz 中心频率下,BP-NGD POC 的 NGD 值为 -4 ns,说明共振效应降低,平整度为 1 dB,从而验证了 EMC 解决方案。此外,时域信号完整性(SI)分析通过显示输出延迟、过射/欠射减少以及输入输出交叉相关性从 89% 提高到 99%,证实了 EMC 谐振腔解决方案。
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引用次数: 0
Design of Perfectly Conducting Objects That Are Invisible to an Incident Plane Wave 设计对入射平面波不可见的完美导电物体
IF 2.3 Q1 Mathematics Pub Date : 2024-02-08 DOI: 10.1109/JMMCT.2024.3364084
Johan Helsing;Shidong Jiang;Anders Karlsson
This work concerns the design of perfectly conducting objects that are invisible to an incident transverse magnetic plane wave. The object in question is a finite planar waveguide with a finite periodic array of barriers. By optimizing this array, the amplitude of the scattered field is reduced to less than $10^{-9}$ times the amplitude of the incident plane wave everywhere outside the waveguide. To accurately evaluate such minute amplitudes, we employ a recently developed boundary integral equation technique, adapted for objects whose boundaries have endpoints, corners, and branch points.
这项工作涉及设计完全导电的物体,使其对入射的横向磁平面波不可见。该物体是一个有限平面波导,带有一个有限周期的屏障阵列。通过优化这个阵列,散射场的振幅会减小到波导外各处入射平面波振幅的 10^{-9}$ 倍以下。为了准确评估这种微小的振幅,我们采用了最近开发的边界积分方程技术,该技术适用于边界有端点、拐角和分支点的物体。
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引用次数: 0
A Fast Iterative Physical Optics Method With Quadratic Amplitude and Phase Integral Terms 带有二次振幅和相位积分项的快速迭代物理光学方法
IF 2.3 Q1 Mathematics Pub Date : 2024-01-25 DOI: 10.1109/JMMCT.2024.3358327
Yang Su;Yu Mao Wu
The iterative physical optics (IPO) method is a valuable technique for analyzing coupled scattering problems. In contrast to the fast physical optics (FPO) method, this article proposes an iterative physical optics method based on quadratic quadrilateral patches (QIPO). Specifically, quadratic patches in the QIPO method offer higher-order accuracy in calculating normal vectors which greatly benefits the accuracy of the iterative induction current. Then, a lit-shadow judgment criterion is introduced, and a general iteration formulation for proposed method is presented. Additionally, new amplitude and phase function expressions suitable for the QIPO method are proposed to accurately compute the far-field results. It is also verified for the case of discretization with quadratic triangular patches (QTIPO). To address numerical singularities, the QIPO method considers a linear phase function, where closed-form solution are provided. The results demonstrate the effectiveness of the treatment in handling singular cases. The accuracy of the QIPO method is validated through comparisons with existing results. Finally, numerical examples confirm that the proposed method reduces the number of patches, minimizes the computational cost of induced current iteration, and accurately calculates far-field results.
迭代物理光学(IPO)方法是分析耦合散射问题的重要技术。与快速物理光学(FPO)方法相比,本文提出了一种基于二次四边形补丁(QIPO)的迭代物理光学方法。具体来说,QIPO 方法中的二次方补丁在计算法向量时提供了更高阶的精度,从而大大提高了迭代感应电流的精度。然后,引入了光影判断标准,并提出了拟议方法的一般迭代公式。此外,还提出了适合 QIPO 方法的新振幅和相位函数表达式,以精确计算远场结果。此外,还验证了使用二次三角形补丁(QTIPO)离散化的情况。为了解决数值奇异性问题,QIPO 方法考虑了线性相位函数,并提供了闭式解。结果证明了该方法在处理奇异情况时的有效性。通过与现有结果的比较,QIPO 方法的准确性得到了验证。最后,数值示例证实,所提出的方法减少了补丁的数量,最大限度地降低了诱导电流迭代的计算成本,并能精确计算远场结果。
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引用次数: 0
Fast Analysis of Broadband Electromagnetic Scattering Problems by Combining Hyper Basis Functions-Based MoM With Compressive Sensing 将基于超基元函数的模拟模型与压缩传感相结合,快速分析宽带电磁散射问题
IF 2.3 Q1 Mathematics Pub Date : 2024-01-23 DOI: 10.1109/JMMCT.2024.3355976
Zhonggen Wang;Chenwei Li;Yufa Sun;Wenyan Nie;Xuejun Zhang;Pan Wang
The hyper basis functions (HBF)-based MoM has been proven to be an efficient numerical method to analyze broadband electromagnetic scattering problems. However, this method costs a lot of time to reconstruct the impedance matrix and reduced matrix at each frequency point. In order to solve the above problem, a novel method combining HBF-based MoM and compressive sensing (CS) has been proposed in this paper. The proposed method first applies the characteristic modes (CM) derived at the highest frequency point as the HBF for solving the scattering problems at lower frequency points, and performs sparse transform of the induced currents as the sparse basis for the CS framework. Then the measurement matrix is constructed using the method of uniformly extracting the impedance matrix by rows to obtain stable calculation results. Finally, according to the prior condition that a few CM are sufficient to characterize the surface currents approximately, the recovery algorithm is simplified least square method to reconstruct the current coefficients. Numerical simulation results show that it can significantly improve the efficiency of solving broadband electromagnetic problems compared with HBF-based MoM.
基于超基函数(HBF)的 MoM 已被证明是一种分析宽带电磁散射问题的高效数值方法。然而,这种方法需要花费大量时间来重建每个频点的阻抗矩阵和还原矩阵。为了解决上述问题,本文提出了一种结合基于 HBF 的 MoM 和压缩传感(CS)的新方法。该方法首先应用在最高频率点得到的特征模态(CM)作为 HBF 来解决较低频率点的散射问题,并对感应电流进行稀疏变换作为 CS 框架的稀疏基础。然后采用按行均匀提取阻抗矩阵的方法构建测量矩阵,从而获得稳定的计算结果。最后,根据几个 CM 就足以近似表征表面电流的先验条件,采用简化的最小二乘法来重建电流系数的恢复算法。数值模拟结果表明,与基于 HBF 的 MoM 相比,它能显著提高解决宽带电磁问题的效率。
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引用次数: 0
Analysis of Q-Factor for AM-SLM Cavity Based Resonators Using Surface Roughness Models 利用表面粗糙度模型分析基于 AM-SLM 腔的谐振器的 Q 系数
IF 2.3 Q1 Mathematics Pub Date : 2024-01-16 DOI: 10.1109/JMMCT.2024.3354489
Qazi Mashaal Khan;Dan Kuylenstierna
This research delves into losses of X-band cavity resonators manufactured using additive manufacturing-selective laser melting (AM-SLM) compared to the standard subtractive manufacturing milling technology. Measured losses are benchmarked in terms of resonator (quality) $Q$-factor. The measured data is further modelled using the Groiss and one-ball Huray models taking into account the implications of surface roughness and electrical conductivity. The unloaded $Q$-factor is derived from frequency-dependent scattering ($S$) parameters obtained from measurements and full-wave simulations. Surface roughness was found to impact the $Q$-factor significantly and the resonant frequency marginally. Cavities based on AM-SLM technology exhibit higher roughness compared to milling and lowers the $Q$-factor. A fusion of both manufacturing methods by milling AM-SLM cavity walls demonstrates an augmented $Q$-factor compared to a directly printed cavity. In the study it was also found that the Groiss model tends to overestimate the $Q$-factor owing to AM-SLM's rougher surface, while the one-ball Huray model furnishes precise projections by establishing a link between surface roughness and powder particles. Electrical conductivity's influence on $Q$-factor was also investigated, showing negligible impact with increased surface roughness. Further, side walls of the AM-SLM cavity were more susceptible to surface roughness, compared to the cavity front walls due to higher surface current density. This study underscores the significance of analyzing surface roughness and electrical conductivity in AM-SLM cavity resonators and highlights the suitability of the one-ball Huray model for accurate $Q$-factor prediction of microwave structures with rough surfaces.
与标准的减法制造铣削技术相比,本研究深入探讨了使用增材制造-选择性激光熔化(AM-SLM)技术制造的 X 波段空腔谐振器的损耗。测量损耗以谐振器(质量)Q$系数为基准。考虑到表面粗糙度和电导率的影响,使用 Groiss 模型和单球 Huray 模型对测量数据进行了进一步建模。根据测量和全波模拟获得的与频率相关的散射 ($S$) 参数推导出空载 $Q$ 因子。研究发现,表面粗糙度对 Q$ 因子的影响很大,而对谐振频率的影响很小。与铣削相比,基于 AM-SLM 技术的腔体表现出更高的粗糙度,并降低了 Q$ 因子。与直接印刷的腔体相比,通过铣削 AM-SLM 腔体壁将两种制造方法融合在一起,显示出更高的 Q$ 因子。研究还发现,由于 AM-SLM 的表面更粗糙,格罗伊斯模型往往会高估 Q 值系数,而单球 Huray 模型则通过建立表面粗糙度和粉末颗粒之间的联系,提供了精确的预测。电导率对 Q$ 因子的影响也进行了研究,结果表明随着表面粗糙度的增加,电导率的影响可以忽略不计。此外,由于表面电流密度较高,与空腔前壁相比,AM-SLM 空腔的侧壁更容易受到表面粗糙度的影响。这项研究强调了分析 AM-SLM 谐振器表面粗糙度和电导率的重要性,并突出了单球 Huray 模型适用于精确预测具有粗糙表面的微波结构的 Q$ 因子。
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引用次数: 0
Maxwell-Schrödinger Modeling of a Superconducting Qubit Coupled to a Transmission Line Network 耦合到传输线网络的超导微ubit 的麦克斯韦-薛定谔模型
IF 2.3 Q1 Mathematics Pub Date : 2024-01-03 DOI: 10.1109/JMMCT.2024.3349433
Thomas E. Roth;Samuel T. Elkin
In superconducting circuit quantum information technologies, classical microwave pulses are applied to control and measure the qubit states. Currently, the design of these microwave pulses uses simple theoretical or numerical models that do not account for the self-consistent interactions of how the qubit state modifies the applied microwave pulse. In this work, we present the formulation and finite element time domain discretization of a semiclassical Maxwell-Schrödinger method for describing these self-consistent dynamics for the case of a superconducting qubit capacitively coupled to a general transmission line network. We validate the proposed method by characterizing key effects related to common control and measurement approaches for transmon and fluxonium qubits in systems that are amenable to theoretical analysis. Our numerical results also highlight scenarios where including the self-consistent interactions is essential. By treating the microwaves classically, our method is substantially more efficient than fully-quantum methods for the many situations where the quantum statistics of the microwaves are not needed. Further, our approach does not require any reformulations when the transmission line system is modified. In the future, our method can be used to rapidly explore broader design spaces to search for more effective control and measurement protocols for superconducting qubits.
在超导电路量子信息技术中,经典微波脉冲用于控制和测量量子比特状态。目前,这些微波脉冲的设计采用简单的理论或数值模型,无法解释量子比特状态如何改变所应用的微波脉冲的自洽相互作用。在这项工作中,我们提出了半经典麦克斯韦-薛定谔方法的公式和有限元时域离散化,用于描述超导量子比特电容耦合到一般传输线网络的自洽动力学。我们通过描述与可进行理论分析的系统中传子和通子量子比特的常见控制和测量方法相关的关键效应,验证了所提出的方法。我们的数值结果还凸显了包含自洽相互作用至关重要的情况。通过对微波进行经典处理,在许多不需要微波量子统计的情况下,我们的方法比全量子方法更有效。此外,当传输线系统被修改时,我们的方法不需要任何重新计算。未来,我们的方法可用于快速探索更广阔的设计空间,为超导量子比特寻找更有效的控制和测量协议。
{"title":"Maxwell-Schrödinger Modeling of a Superconducting Qubit Coupled to a Transmission Line Network","authors":"Thomas E. Roth;Samuel T. Elkin","doi":"10.1109/JMMCT.2024.3349433","DOIUrl":"https://doi.org/10.1109/JMMCT.2024.3349433","url":null,"abstract":"In superconducting circuit quantum information technologies, classical microwave pulses are applied to control and measure the qubit states. Currently, the design of these microwave pulses uses simple theoretical or numerical models that do not account for the self-consistent interactions of how the qubit state modifies the applied microwave pulse. In this work, we present the formulation and finite element time domain discretization of a semiclassical Maxwell-Schrödinger method for describing these self-consistent dynamics for the case of a superconducting qubit capacitively coupled to a general transmission line network. We validate the proposed method by characterizing key effects related to common control and measurement approaches for transmon and fluxonium qubits in systems that are amenable to theoretical analysis. Our numerical results also highlight scenarios where including the self-consistent interactions is essential. By treating the microwaves classically, our method is substantially more efficient than fully-quantum methods for the many situations where the quantum statistics of the microwaves are not needed. Further, our approach does not require any reformulations when the transmission line system is modified. In the future, our method can be used to rapidly explore broader design spaces to search for more effective control and measurement protocols for superconducting qubits.","PeriodicalId":52176,"journal":{"name":"IEEE Journal on Multiscale and Multiphysics Computational Techniques","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139473800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
PIFON-EPT: MR-Based Electrical Property Tomography Using Physics-Informed Fourier Networks PIFON-EPT:利用物理信息傅立叶网络进行基于磁共振的电特性断层扫描
IF 2.3 Q1 Mathematics Pub Date : 2023-12-22 DOI: 10.1109/JMMCT.2023.3345798
Xinling Yu;José E. C. Serrallés;Ilias I. Giannakopoulos;Ziyue Liu;Luca Daniel;Riccardo Lattanzi;Zheng Zhang
We propose Physics-Informed Fourier Networks for Electrical Properties (EP) Tomography (PIFON-EPT), a novel deep learning-based method for EP reconstruction using noisy and/or incomplete magnetic resonance (MR) measurements. Our approach leverages the Helmholtz equation to constrain two networks, responsible for the denoising and completion of the transmit fields, and the estimation of the object's EP, respectively. We embed a random Fourier features mapping into our networks to enable efficient learning of high-frequency details encoded in the transmit fields. We demonstrated the efficacy of PIFON-EPT through several simulated experiments at 3 and 7 T (T) MR imaging, and showed that our method can reconstruct physically consistent EP and transmit fields. Specifically, when only 20% of the noisy measured fields were used as inputs, PIFON-EPT reconstructed the EP of a phantom with $leq 5%$ error, and denoised and completed the measurements with $leq 1%$ error. Additionally, we adapted PIFON-EPT to solve the generalized Helmholtz equation that accounts for gradients of EP between inhomogeneities. This yielded improved results at interfaces between different materials without explicit knowledge of boundary conditions. PIFON-EPT is the first method that can simultaneously reconstruct EP and transmit fields from incomplete noisy MR measurements, providing new opportunities for EPT research.
我们提出了用于电特性(EP)断层扫描的物理信息傅立叶网络(PIFON-EPT),这是一种基于深度学习的新方法,用于利用有噪声和/或不完整的磁共振(MR)测量结果重建电特性。我们的方法利用亥姆霍兹方程来约束两个网络,分别负责传输场的去噪和补全,以及对象 EP 的估计。我们在网络中嵌入了随机傅立叶特征映射,以便高效学习发射场中的高频细节编码。我们通过 3 T 和 7 T 磁共振成像的几个模拟实验证明了 PIFON-EPT 的功效,并表明我们的方法可以重建物理上一致的 EP 和发射场。具体来说,当只有20%的噪声测量场被用作输入时,PIFON-EPT重建的幻影EP误差为5%,去噪并完成测量的误差为1%。此外,我们对 PIFON-EPT 进行了调整,以求解广义亥姆霍兹方程,该方程考虑了非均质间 EP 的梯度。这改进了不同材料界面的结果,而无需明确了解边界条件。PIFON-EPT 是第一种能从不完整的噪声磁共振测量中同时重建 EP 和透射场的方法,为 EPT 研究提供了新的机遇。
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引用次数: 0
Enhanced Thermodynamic Modeling of Converter Transformer Influenced by DC Bias 受直流偏置影响的变流器变压器的强化热力学建模
IF 2.3 Q1 Mathematics Pub Date : 2023-11-20 DOI: 10.1109/JMMCT.2023.3334563
Suman Yadav;Gourav Kumar Suman;Ram Krishna Mehta
DC bias in high-voltage DC transformers, arising from converter operations and DC transmission, poses significant challenges to their performance. The detrimental effects of DC bias primarily manifest in increased temperature, jeopardizing the safe operation of the transformers. This article presents a novel approach by extending the utilization of the Thermal Equivalent Circuit (TEC) to accurately predict temperatures at different elements of a converter transformer under DC bias conditions. Specifically designed for a 240 MVA converter transformer, the TEC incorporates capacitances and dynamic resistances as model parameters. Additionally, an electro-thermal finite element model is implemented to comprehensively analyze the transformer's behavior under varying levels of DC bias. To estimate the TEC parameters, a hybrid GWO-CS (Grey Wolf Optimization – Cuckoo Search) algorithm is employed based on measured values. Furthermore, the paper highlights the impact of DC bias on the converter transformer's life expectancy, considering the aging acceleration factor.
高压直流变压器中的直流偏压是由变流器运行和直流输电引起的,这对变压器的性能提出了重大挑战。直流偏压的有害影响主要表现为温度升高,危及变压器的安全运行。本文通过扩展热等效电路 (TEC) 的使用范围,提出了一种新方法,以准确预测直流偏置条件下换流器变压器不同元件的温度。热等效电路专为 240 MVA 变流器变压器设计,将电容和动态电阻作为模型参数。此外,还采用了电热有限元模型,以全面分析变压器在不同直流偏置水平下的行为。为了估算 TEC 参数,采用了基于测量值的混合 GWO-CS(灰狼优化-布谷鸟搜索)算法。此外,考虑到老化加速因素,本文还强调了直流偏压对转换变压器预期寿命的影响。
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引用次数: 0
Multiphysics Analysis of a High-Speed Eddy Current Brake 高速涡流制动器的多物理场分析
IF 2.3 Q1 Mathematics Pub Date : 2023-11-16 DOI: 10.1109/JMMCT.2023.3333386
Sandeep Mohan Nayak;Mangal Kothari;Abhishek Sarkar;Soumya Ranjan Sahoo
This paper presents the analytical-cum-numerical-based mathematical model for the multiphysics simulation of a high-speed unipolar axial eddy current brake (ECB). The operating principle and the necessary multiphysics simulation for an ECB are introduced. An analytical method is developed for high-speed ECB operation to search coarse parameters in the preliminary design technique. The model uses the behavior of the eddy currents on the plate during high-speed operation. A radial multiplier is incorporated to satisfy electromagnetic physics. The axisymmetric property of the disk reduces the disk geometry to an equivalent 2D domain where the estimated loss is defined. The ohmic loss from the analytical model is transferred to the numerical thermal model to evaluate temperature distribution. The convective heat transfer coefficient, which is a crucial variable in boundary conditions, is defined using the correlations between the Nusselt and Reynolds numbers. The steady-state heat diffusion equation is solved in the domain for three different speeds. The results show that the ohmic loss on the disk saturates and the temperature of the disk reduces during high-speed operations.
本文建立了高速单极轴向涡流制动器多物理场仿真的解析与数值相结合的数学模型。介绍了ECB的工作原理和必要的多物理场仿真。在初步设计技术中,提出了一种用于高速ECB运行的粗参数搜索的解析方法。该模型采用了高速运行时板上涡流的行为。采用径向乘法器满足电磁物理要求。磁盘的轴对称特性将磁盘几何形状降低到等效的二维域,其中定义了估计损耗。将解析模型的欧姆损耗转换为数值热模型来计算温度分布。对流换热系数是边界条件下的一个重要变量,利用努塞尔数和雷诺数之间的相关性来定义对流换热系数。在三种不同速度下,求解了稳态热扩散方程。结果表明,在高速运行时,磁盘上的欧姆损耗趋于饱和,磁盘温度降低。
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引用次数: 0
Assessing Curl-Conforming Bases for Pyramid Cells 评估金字塔细胞的卷发符合基
IF 2.3 Q1 Mathematics Pub Date : 2023-11-16 DOI: 10.1109/JMMCT.2023.3333563
Roberto D. Graglia;Paolo Petrini
Successful three-dimensional finite element codes for Maxwell's equations must include and deal with all four types of geometrical shapes: tetrahedra, bricks, prisms, and quadrangular-based pyramids. However, pyramidal elements have so far been used very rarely because the basis functions associated with them have complicated expression, are complex in derivation, and have never been comprehensively validated. We recently published a simpler procedure for constructing higher-order vector bases for pyramid elements, so here we fill a gap by discussing a whole set of test case results that not only validate our new curl-conforming bases for pyramids, but which enable validation of other codes that use pyramidal elements for finite element method applications. The solutions of the various test cases are obtained using either higher order elements or multipyramidal meshes or both. Furthermore, the results are always compared with the solutions obtained with classical tetrahedral meshes using higher order bases. This allows us to verify that purely pyramidal meshes and elements give numerical results of comparable accuracy to those obtained with multitetrahedral meshes that use elements of the same order, essentially requiring the same number of degrees of freedom. The various results provided here also show that higher order vector bases always guarantee a superior convergence of the numerical results as the number of degrees of freedom increases.
麦克斯韦方程组的成功三维有限元代码必须包括并处理所有四种几何形状:四面体、砖块、棱镜和四边形金字塔。然而,由于与之相关的基函数表达式复杂,推导过程复杂,而且尚未得到全面的验证,金字塔元迄今很少被使用。我们最近发布了一个为金字塔单元构建高阶向量基的更简单的过程,因此在这里我们通过讨论一整套测试用例结果来填补空白,这些结果不仅验证了我们新的符合螺旋形的金字塔基,而且还验证了其他使用金字塔单元进行有限元方法应用的代码。各种测试用例的解可采用高阶元或多锥体网格,或两者兼而有之。此外,结果总是与使用高阶基的经典四面体网格的解进行比较。这使我们能够验证,纯锥体网格和单元给出的数值结果与使用相同顺序的元素的多四面体网格获得的结果相当,基本上需要相同数量的自由度。这里提供的各种结果也表明,随着自由度的增加,高阶向量基总是保证数值结果的优越收敛性。
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引用次数: 0
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IEEE Journal on Multiscale and Multiphysics Computational Techniques
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