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Geophysicists today face the challenge of quickly and reliably interpreting extensive controlled-source electromagnetic (CSEM) data sets to map subsurface conductivity structures within realistic geological environments. An ideal 3-D CSEM inversion algorithm using tetrahedral grids should be capable of distinguishing different resolution requirements between forward modelling and inversion grids, have an optimal parallel strategy that fully exploits the inherent independence of CSEM data sets while also possessing the capability to handle large-scale geo-electrical models, and incorporate conductivity anisotropy which should be a common characteristic in realistic subsurface environments. However, existing tools in the geo-electromagnetic community often fall short of these three demands. Addressing this gap, our study introduces a scalable and parallel anisotropic inversion technique for CSEM data, capitalizing on the potential of unstructured tetrahedral grids. We first apply the tetrahedral longest-edge bisection method to create a refined dense, heterogeneous forward modelling grid from a coarse inversion grid. This refinement, focused on areas around transmitters and receivers, is seamlessly integrated within the coarser inversion grid's topology, enabling precise conductivity mapping and preserving electromagnetic response accuracy during model updates. We further innovate with a source-mesh double-level parallel strategy, utilizing the message passing interface technique for parallel handling of independent CSEM data sets and large-scale geo-electrical models. Externally, we dedicate a processor for inversion model updates employing the Limited-memory Broyden–Fletcher–Goldfarb–Shanno optimization algorithm and divide other processors into groups, each associated with specific transmitting sources and frequencies. Internally, in each group, we employ a domain-decomposition-based scalable and robust iterative solvers using the Auxiliary-Space Maxwell pre-conditioner to parallel quickly calculate the electromagnetic responses from its assigned source-frequency set. Additionally, recognizing the potential for electrical conductivity anisotropy in field data, we incorporate the case of vertical transverse isotropy. We validate the effectiveness of our method through examples, including an isotropic land model with undulating topography, an anisotropic marine model and a real-field data case. Results from both synthetic and field data inversions underscore our method's significant advancements in efficiency and practicality, particularly in addressing large-scale 3-D CSEM data sets inversion challenges in realistic geological environments. [ABSTRACT FROM AUTHOR]

This paper aims to obtain solutions in terms of the complex potential structure for the plain strain problem of an elastic micropolar and isotropic body with pores. The constitutive equations on which the method is applied and is useful are the well-known equations of the elasticity theory for the above-mentioned body. We intend to solve the Kirch problem using the procedure of complex variables. [ABSTRACT FROM AUTHOR]

Nonlinear problems, which often arise in various scientific and engineering disciplines, typically involve nonlinear equations or functions with multiple solutions. Analytical solutions to these problems are often impossible to obtain, necessitating the use of numerical techniques. This research proposes an efficient and stable Caputo-type inverse numerical fractional scheme for simultaneously approximating all roots of nonlinear equations, with a convergence order of 2 ψ + 2 . The scheme is applied to various nonlinear problems, utilizing dynamical analysis to determine efficient initial values for a single root-finding Caputo-type fractional scheme, which is further employed in inverse fractional parallel schemes to accelerate convergence rates. Several sets of random initial vectors demonstrate the global convergence behavior of the proposed method. The newly developed scheme outperforms existing methods in terms of accuracy, consistency, validation, computational CPU time, residual error, and stability. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive review of research conducted on the Method of Auxiliary Sources (MAS) over a period of the last 22 years, i.e., since the last up-to-date survey was published. MAS is a very attractive numerical technique due to its simple algorithmic structure and the generally low computational cost it requires in terms of memory and CPU time; this is why it has been applied to a vast variety of cases, as concluded by the long citations list included. After a short introduction summarizing the fundamental concepts of the method, references since 2002 are categorized, briefly described, and commented on. This work is intended to assist every researcher who is involved in MAS computations, providing an exhaustive, to the best of the authors' knowledge, list of related publications. [ABSTRACT FROM AUTHOR]

We study symmetries of bases and spanning sets in finite element exterior calculus, using representation theory. We want to know which vector-valued finite element spaces have bases invariant under permutation of vertex indices. The permutations of vertex indices correspond to the symmetry group of the simplex. That symmetry group is represented on simplicial finite element spaces by the pullback action. We determine a natural notion of invariance and sufficient conditions on the dimension and polynomial degree for the existence of invariant bases. We conjecture that these conditions are necessary too. We utilize Djoković and Malzan's classification of monomial irreducible representations of the symmetric group and show new symmetries of the geometric decomposition and canonical isomorphisms of the finite element spaces. Explicit invariant bases with complex coefficients are constructed in dimensions two and three for different spaces of finite element differential forms. [ABSTRACT FROM AUTHOR]

In this paper, we study the improved block splitting (IBS) iteration method and its accelerated variant, the accelerated improved block splitting (AIBS) iteration method, for solving linear systems of equations stemming from the discretization of the complex Helmholtz equation. We conduct a comprehensive convergence analysis and derive optimal iteration parameters aimed at minimizing the spectral radius of the iteration matrix. Through numerical experiments, we validate the efficiency of both iteration methods. [ABSTRACT FROM AUTHOR]

This work develops optimal preconditioners for the discrete H(curl) and H(div) problems on two-dimensional surfaces by nodal auxiliary space preconditioning (Hiptmair and Xu in SIAM J Numer Anal 45:2483–2509, 2007). In particular, on unstructured triangulated surfaces, we develop fast and user-friendly preconditioners for the edge and face element discretizations of curl–curl and grad–div problems based on inverting several discrete surface Laplacians. The proposed preconditioners lead to efficient iterative methods for computing harmonic tangential vector fields on discrete surfaces. Numerical experiments on two- and three-dimensional hypersurfaces are presented to test the performance of those surface preconditioners. [ABSTRACT FROM AUTHOR]

Being able to characterise objects at low frequencies, but in situations where the modelling error in the eddy current approximation of the Maxwell system becomes large, is important for improving current metal detection technologies. Importantly, the modelling error becomes large as the frequency increases, but the accuracy of the eddy current model also depends on the object topology and on its materials, with the error being much larger for certain geometries compared to others of the same size and materials. Additionally, the eddy current model breaks down at much smaller frequencies for highly magnetic conducting materials compared to non-permeable objects (with similar conductivities, sizes and shapes) and, hence, characterising small magnetic objects made of permeable materials using the eddy current at typical frequencies of operation for a metal detector is not always possible. To address this, we derive a new asymptotic expansion for permeable highly conducting objects that is valid for small objects and holds not only for frequencies where the eddy current model is valid but also for situations where the eddy current modelling error becomes large and applying the eddy approximation would be invalid. The leading-order term we derive leads to new forms of object characterisations in terms of polarizability tensor object descriptions where the coefficients can be obtained from solving vectorial transmission problems. We expect these new characterisations to be important when considering objects at greater stand-off distance from the coils, which is important for safety critical applications, such as the identification of landmines, unexploded ordnance and concealed weapons. We also expect our results to be important when characterising artefacts of archaeological and forensic significance at greater depths than the eddy current model allows and to have further applications parking sensors and improving the detection of hidden, out-of-sight, metallic objects. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical solution of a large complex linear system with a saddle-point form obtained by the discretization of the time-harmonic eddy-current optimal control problem. A new Schur complement is proposed for this algebraic system, extending it to both the block-triangular preconditioner and the structured preconditioner. A theoretical analysis proves that the eigenvalues of block-triangular and structured preconditioned matrices are located in the interval [1/2, 1]. Numerical simulations show that two new preconditioners coupled with a Krylov subspace acceleration have good feasibility and effectiveness and are superior to some existing efficient algorithms. [ABSTRACT FROM AUTHOR]

This work deals with the spectral element discretization of the time-dependent Stokes problem in two- and three-dimensional domains. The boundary condition is defined on the normal component of the velocity and the tangential components of the vorticity. The discretization related to the time variable is processed by a Backward Euler method. We prove through a detailed numerical analysis the well-posedness of the full discrete problem. [ABSTRACT FROM AUTHOR]

Geomagnetic depth sounding (GDS) is a geophysical electromagnetic (EM) method that studies the deep structure and composition of the Earth by using long-period EM signals from geomagnetic observatories and satellites. In this paper, a 3-D anisotropic GDS modelling algorithm is developed. The curl–curl equation is discretized using the edge-based finite-element method on unstructured tetrahedral grids. In order to solve the computationally demanding problem of EM modelling on a global scale, the complex linear system is first separated into the equivalent real linear systems and then the real system is iteratively solved by the flexible generalized minimum residual method with a block diagonal pre-conditioner. This will greatly reduce the condition number of the linear system and thus speed up the solution process. We verify the accuracy of the proposed algorithm by comparing our results with the existing methods. After that, we design a subduction zone model to simulate the EM responses under isotropic and anisotropic environments, respectively. The numerical results show the high efficiency of the proposed algorithm and the response differences between isotropic and anisotropic models. This research can provide theoretical and technical support for the high-accuracy and efficient inversion of GDS data for the geo-dynamic study. [ABSTRACT FROM AUTHOR]

For the incompressible MHD equations, we present a fully discrete second-order-in-time scheme based on a blended BDF and extrapolation treatments for nonlinear terms. The proposed scheme is more accurate than the two-step BDF with additional nominal computational time and is still A-stable. Then, unconditional stability, long-time stability, and optimal convergence rate of the scheme are presented. At last, the numerical findings are verified by comparison with the two-step BDF scheme in several 2D and 3D numerical experiments. [ABSTRACT FROM AUTHOR]

In this article, we present algorithms and implementations for the end-to-end GPU acceleration of matrix-free low-order-refined preconditioning of high-order finite element problems. The methods described here allow for the construction of effective preconditioners for high-order problems with optimal memory usage and computational complexity. The preconditioners are based on the construction of a spectrally equivalent low-order discretization on a refined mesh, which is then amenable to, for example, algebraic multigrid preconditioning. The constants of equivalence are independent of mesh size and polynomial degree. For vector finite element problems in H (curl) and H (div) (e.g., for electromagnetic or radiation diffusion problems), a specially constructed interpolation–histopolation basis is used to ensure fast convergence. Detailed performance studies are carried out to analyze the efficiency of the GPU algorithms. The kernel throughput of each of the main algorithmic components is measured, and the strong and weak parallel scalability of the methods is demonstrated. The different relative weighting and significance of the algorithmic components on GPUs and CPUs is discussed. Results on problems involving adaptively refined nonconforming meshes are shown, and the use of the preconditioners on a large-scale magnetic diffusion problem using all spaces of the finite element de Rham complex is illustrated. [ABSTRACT FROM AUTHOR]

Abnormal formation of girth weld is a major threat to the safe operation of pipelines, which may lead to serious accidents. Therefore, regular inspection and maintenance of girth weld are essential for accident prevention and energy security. This paper presents a novel method for inspecting abnormal girth weld formation in oil and gas pipelines using alternating excitation detection technology. The method is based on the analysis of the microscopic magnetic variations in the welded area under alternating magnetic fields. An internal inspection probe and electronic system for detecting abnormal girth weld formation were designed and developed. The system's capability to identify misalignment, undercutting, root concavity, and abnormal formation height of girth weld was tested by numerical simulation and experimental study. The results show that the detection system can effectively identify a minimum misalignment of 0.5 mm at a lift-off height of 15 mm. The proposed method offers several advantages, such as rapid response, low cost, non-contact operation, and high sensitivity to surface flaws in ferromagnetic pipelines. [ABSTRACT FROM AUTHOR]

We develop a theory of finite element systems, for the purpose of discretizing sections of vector bundles, in particular those arising in the theory of elasticity. In the presence of curvature, we prove a discrete Bianchi identity. In the flat case, we prove a de Rham theorem on cohomology groups. We check that some known mixed finite elements for the stress–displacement formulation of elasticity fit our framework. We also define, in dimension two, the first conforming finite element spaces of metrics with good linearized curvature, corresponding to strain tensors with Saint-Venant compatibility conditions. Cochains with coefficients in rigid motions are given a key role in relating continuous and discrete elasticity complexes. [ABSTRACT FROM AUTHOR]

We present a novel a , v -q hybrid method for solving large-scale time-harmonic eddy-current problems. This method combines a hybrid unsymmetric formulation based on the cell method and the boundary element method with a hierarchical matrix-compression technique based on randomized singular value decomposition. The main advantage is that the memory requirements are strongly reduced compared to the corresponding hybrid method without matrix compression while retaining the same robust solution strategy consisting of a simple construction of the preconditioner. In addition, the matrix-compression accuracy and efficiency are enhanced compared to traditional compression methods, such as adaptive cross approximation. The numerical results show that the proposed hybrid approach can also be effectively used to analyze large-scale eddy-current problems of engineering interest. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel arbitrary-order discrete de Rham (DDR) complex on general polyhedral meshes based on the decomposition of polynomial spaces into ranges of vector calculus operators and complements linked to the spaces in the Koszul complex. The DDR complex is fully discrete, meaning that both the spaces and discrete calculus operators are replaced by discrete counterparts, and satisfies suitable exactness properties depending on the topology of the domain. In conjunction with bespoke discrete counterparts of L 2 -products, it can be used to design schemes for partial differential equations that benefit from the exactness of the sequence but, unlike classical (e.g., Raviart–Thomas–Nédélec) finite elements, are nonconforming. We prove a complete panel of results for the analysis of such schemes: exactness properties, uniform Poincaré inequalities, as well as primal and adjoint consistency. We also show how this DDR complex enables the design of a numerical scheme for a magnetostatics problem, and use the aforementioned results to prove stability and optimal error estimates for this scheme. [ABSTRACT FROM AUTHOR]

This article focuses on the development of error estimators for evaluating the accuracy of finite-element computations of the electromagnetic phenomena for electromagnetic material processing modeling. The ultimate goal of this work is to enable adaptive anisotropic remeshing in order to reach a prescribed accuracy. We first introduce a numerical model based on the quasi-steady-state approximation of the Maxwell’s equations in a time-domain formalism. We then introduce an estimator based on a recovery method; its results will be compared with other estimators based on investigating how accurately some of the electromagnetic equations are checked. The estimators are then validated on analytical cases; an additional application to a complex case shows the robustness of our approach. Scientific advances of this work include the development of a robust field recovery strategy for error computation, as well as the introduction of a general purpose error normalization scheme which overcomes field scale bias enabling seamless comparison of errors from different physical fields. [ABSTRACT FROM AUTHOR]

Physics-informed neural networks approximate solutions of PDEs by minimizing pointwise residuals. We derive rigorous bounds on the error, incurred by PINNs in approximating the solutions of a large class of linear parabolic PDEs, namely Kolmogorov equations that include the heat equation and Black-Scholes equation of option pricing, as examples. We construct neural networks, whose PINN residual (generalization error) can be made as small as desired. We also prove that the total L2-error can be bounded by the generalization error, which in turn is bounded in terms of the training error, provided that a sufficient number of randomly chosen training (collocation) points is used. Moreover, we prove that the size of the PINNs and the number of training samples only grow polynomially with the underlying dimension, enabling PINNs to overcome the curse of dimensionality in this context. These results enable us to provide a comprehensive error analysis for PINNs in approximating Kolmogorov PDEs. [ABSTRACT FROM AUTHOR]

The fast and accurate 3-D magnetotelluric (MT) forward modelling is core engine of the interpretation and inversion of MT data. In this study, we develop an improved extrapolation cascadic multigrid method (EXCMG) to solve the large sparse complex linear system arising from the finite-element (FE) discretization on non-uniform orthogonal grids of the Maxwell's equations using potentials. First, the vector Helmholtz equation and the scalar auxiliary equation are derived from the Maxwell's equations using Coulomb-gauged potentials. The weighted residual method is adopted to discretize the weak formulation and assemble the FE equation. Secondly, carefully choosing the preconditioned complex stable bi-conjugate gradient method (BiCGStab) as multigrid smoother, we develop an improved EXCMG method on non-uniform grids to solve the resulting large sparse complex non-Hermitian linear systems. Finally, several examples including three standard testing models (COMMEMI3D-1, COMMEMI3D-2 and DTM1.0) and a topographic model are used to validate the accuracy and efficiency of the proposed multigrid solver. Numerical results show that the proposed EXCMG algorithm greatly improves the efficiency of 3-D MT forward modelling, is more efficient than some existing solvers, such as Pardiso, incomplete LU factorization preconditioned biconjugate gradients stabilized method (ILU-BiCGStab) and flexible generalized minimum residual method with auxiliary space Maxwell preconditioner (FGMRES-AMS), and capable to simulate large-scale problems with more than 100 million unknowns. [ABSTRACT FROM AUTHOR]

Geo-electromagnetic forward modeling problems are ill-posed due to the low signal frequencies being used and electrically insulating air space. To overcome this numerical issue, the |$A - \phi $| formula using the magnetic vector potentials (⁠|$\bf A$|⁠) and electric scalar potentials (⁠|$\phi $|⁠) was developed. At present, there are two sets of |$A - \phi $| formulae being used: one has a curl–curl (⁠|$\nabla \times \nabla $|⁠) structure and another one has a Laplace (⁠|${\nabla ^2}$|⁠) structure where the Coulomb gauge is enforced. The question as to which of the two approaches have superior performance for 3D geo-electromagnetic induction problems still remains open. In this study, we systemically compared the performances of these two |$A - \phi $| systems in terms of both numerical accuracy and convergence rate. Numerical experiments suggest that for both magnetotelluric and controlled-source electromagnetic problems, the |$A - \phi $| system with Laplace structure has better performance than the variant with curl–curl structure in terms of convergence rates. [ABSTRACT FROM AUTHOR]

We analyze a new stabilized dual-mixed method applied to incompressible linear elasticity problems, considering two kinds of data on the boundary of the domain: non homogeneous Dirichlet and mixed boundary conditions. In this approach, we circumvent the standard use of the rotation to impose weakly the symmetry of stress tensor. We prove that the new variational formulation and the corresponding Galerkin scheme are well-posed. We also provide the rate of convergence when each row of the stress is approximated by Raviart-Thomas elements and the displacement is approximated by continuous piecewise polynomials. Moreover, we derive a residual a posteriori error estimator for each situation. The corresponding analysis is quite different, depending on the type of boundary conditions. For known displacement on the whole boundary, we based our analysis on Ritz projection of the error, which requires a suitable quasi-Helmholtz decomposition of functions living in H(div;Ω). As a result, we obtain a simple a posteriori error estimator, which consists of five residual terms, and results to be reliable and locally efficient. On the other hand, when we consider mixed boundary conditions, these tools are not necessary. Then, we are able to develop an a posteriori error analysis, which provides us of an estimator consisting of three residual terms. In addition, we prove that in general this estimator is reliable, and when the traction datum is piecewise polynomial, locally efficient. In the second situation, we propose a numerical procedure to compute the numerical approximation, at a reasonable cost. Finally, we include several numerical experiments that illustrate the performance of the corresponding adaptive algorithm for each problem, and support its use in practice. [ABSTRACT FROM AUTHOR]

We consider GMRES applied to discretisations of the high-frequency Helmholtz equation with strong trapping; recall that in this situation the problem is exponentially ill-conditioned through an increasing sequence of frequencies. Our main focus is on boundary-integral-equation formulations of the exterior Dirichlet and Neumann obstacle problems in 2- and 3-d. Under certain assumptions about the distribution of the eigenvalues of the integral operators, we prove upper bounds on how the number of GMRES iterations grows with the frequency; we then investigate numerically the sharpness (in terms of dependence on frequency) of both our bounds and various quantities entering our bounds. This paper is therefore the first comprehensive study of the frequency-dependence of the number of GMRES iterations for Helmholtz boundary-integral equations under trapping. [ABSTRACT FROM AUTHOR]

Direction-of-arrival (DOA) estimation is the preliminary stage of communication, localization, and sensing. Hence, it is a canonical task for next-generation wireless communications, namely beyond 5G (B5G) or 6G communication networks. Both massive multiple-input multiple-output (MIMO) and millimeter wave (mmW) bands are emerging technologies that can be implemented to increase the spectral efficiency of an area, and a number of expectations have been placed on them for future-generation wireless communications. Meanwhile, they also create new challenges for DOA estimation, for instance, through extremely large-scale array data, the coexistence of far-field and near-field sources, mutual coupling effects, and complicated spatial-temporal signal sampling. This article discusses various open issues related to DOA estimation for B5G/6G communication networks. Moreover, some insights on current advances, including arrays, models, sampling, and algorithms, are provided. Finally, directions for future work on the development of DOA estimation are addressed. [ABSTRACT FROM AUTHOR]

In this article, a mixed finite element method for thermally coupled, stationary incompressible MHD problems with physical parameters dependent on temperature in the Lipschitz domain is considered. Due to the variable coefficients of the MHD model, the nonlinearity of the system is increased. A stationary discrete scheme based on the coefficients dependent temperature is proposed, in which the magnetic equation is approximated by Nédélec edge elements, and the thermal and Navier–Stokes equations are approximated by the mixed finite elements. We rigorously establish the optimal error estimates for velocity, pressure, temperature, magnetic induction and Lagrange multiplier with the hypothesis of a low regularity for the exact solution. Finally, a numerical experiment is provided to illustrate the performance and convergence rates of our numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned about a time-domain carpet cloak model, which was originally derived in our previous work Li et al. (SIAM J. Appl. Math., 74(4), pp. 1136–1151, 2014). Some finite element schemes have been developed for this model and used to simulate the cloaking phenomenon in Li et al. (SIAM J. Appl. Math., 74(4), pp. 1136–1151, 2014) and Li et al. (Methods Appl. Math., 19(2), pp. 359–378, 2019). However, numerical stabilities for those proposed explicit schemes are only proved under the time step constrain τ = O(h2), which is impractical and too restricted. To overcome this disadvantage, we propose two new finite element schemes for solving this carpet cloak model: one is the implicit Crank-Nicolson (CN) scheme, and another one is the explicit leap-frog (LF) scheme. Inspired by a totally new energy developed for the continuous model, we prove the unconditional stability for the CN scheme and conditional stability for the LF scheme under the usual CFL constraint τ = O(h). Both numerical stabilities inherit the exact form as the continuous stability. Optimal error estimate is also established for the LF scheme. Finally, numerical results using the LF scheme are presented to support our analysis and demonstrate the cloaking phenomenon. [ABSTRACT FROM AUTHOR]

Parasitic extraction is a powerful tool in the design process of electromechanical devices, specifically as part of workflows that check electromagnetic compatibility. A novel scheme to extract impedances from CAD device models, suitable for a finite element implementation, is derived from Maxwell's equations in differential form. It provides a foundation for parasitic extraction across a broad frequency range and is able to handle inhomogeneous permittivities and permeabilities, making it more flexible than existing integral equation approaches. The approach allows for the automatic treatment of multi-port models of arbitrary conductor geometry without requiring any significant manual user interaction. This is achieved by computing a connecting source current density that supplies current to the model's terminals, whatever their location in the model, subsequently using this current density to compute the electric field, and finally calculating the impedance via a scalar potential. A mandatory low-frequency stabilization scheme is outlined, ensuring a stable evaluation of the model at low frequencies as well. Two quasistatic approximations and the special case of perfect electric conductors are treated theoretically. The magnetoquasistatic approximation is validated against an analytical model in a numerical experiment. Moreover, the intrinsic capability of the method to treat inhomogeneous permittivities and permeabilities is demonstrated with a simple capacitor-coil model including dielectric insulation and magnetic core materials. Finally, the method's practicality is exemplified with a common mode choke model in a comparison of simulated and measured results. [ABSTRACT FROM AUTHOR]

We give a systematic self-contained exposition of how to construct geometrically decomposed bases and degrees of freedom in finite element exterior calculus. In particular, we elaborate upon a previously overlooked basis for one of the families of finite element spaces, which is of interest for implementations. Moreover, we give details for the construction of isomorphisms and duality pairings between finite element spaces. These structural results show, for example, how to transfer linear dependencies between canonical spanning sets, or how to derive the degrees of freedom. [ABSTRACT FROM AUTHOR]

Partition of unity finite element method with plane wave enrichment (PW-FEM) uses a shape function with a set of plane waves propagating in various directions. For room acoustic simulations in a frequency domain, PW-FEM can be an efficient wave-based prediction method, but its practical applications and especially its robustness must be studied further. This study elucidates PW-FEM robustness via 2D real-scale office room problems including rib-type acoustic diffusers. We also demonstrate PW-FEM performance using a sparse direct solver and a high-order Gauss–Legendre rule with a recently developed rule for ascertaining the number of integration points against the classical linear and quadratic FEMs. Numerical experiments investigating mesh size and room geometrical complexity effects on the robustness of PW-FEM demonstrated that PW-FEM becomes more robust at wide bands when using a mesh in which the maximum element size maintains a comparable value to the wavelength of the upper-limit frequency. Moreover, PW-FEM becomes unstable with lower spatial resolution mesh, especially for rooms with complex shape. Comparisons of accuracies and computational costs of linear and quadratic FEM revealed that PW-FEM requires twice the computational time of the quadratic FEM with a mesh having spatial resolution of six elements per wavelength, but it is highly accurate at wide bands with lower memory and with markedly fewer degrees of freedom. As an additional benefit of PW-FEM, the impulse response waveform of quadratic FEM in a time domain was found to deteriorate over time, but the PW-FEM waveform can maintain accurate waveforms over a long time. [ABSTRACT FROM AUTHOR]

We consider a two-part method for computing the acoustic scattering T-matrix of a three dimensional particle. The first part involves accurately computing the far fields by solving a number of particular scattering problems. The second part calculates the T-matrix from these far fields using the Fourier transform over the sphere. The two-part method was first introduced in Ganesh and Hawkins [J. Comput. Appl. Math. 234, 1702–1709]. The focus of this work is to demonstrate the numerical stability and physical correctness of the two-part method for scattering by nonspherical particles with large aspect ratios and size parameters that are at the upper limit of numerical stability for the current state-of-the-art algorithm. The numerical stability of the method is attributed to elimination of the Hankel functions by working with the far field. The numerical experiments use our recently developed open-source software package (TMATROM3) that implements the two-part method. [ABSTRACT FROM AUTHOR]

A natural coupling of a circuit with an electromagnetic device is possible if special boundary conditions, called Electric Circuit Element (ECE), are used for the electromagnetic field formulation. This contribution shows how these ECE boundary conditions can be implemented into the 3D-finite element method for solving coupled full-wave electromagnetic (EM) field-circuit problems in the frequency domain. The frequency response allows the extraction of a reduced order model of the analyzed device, accounting for all the EM field effects. The implementation is based on a weak formulation that uses the electric field strength E strictly inside the domain and a scalar potential V defined solely at the boundary. Edge elements for E are used inside the three-dimensional domain and nodal elements for V are used on its two-dimensional boundary. The weak formulation is described and implemented in the free environment Open Numerical Engineering LABoratory (onelab). The validation is carried out on 3D examples. [ABSTRACT FROM AUTHOR]

Low-frequency sound field reconstruction in an enclosed space has many applications where the plane wave approximation of acoustic modes plays a crucial role. However, the basis mismatch of the plane wave directions degrades the approximation accuracy. In this study, a two-stage method combining ℓ 1 -norm relaxation and parametric sparse Bayesian learning is proposed to address this problem. This method involves selecting sparse dominant plane wave directions from pre-discretized directions and constructing a parameterized dictionary of low dimensionality. This dictionary is used to re-estimate the plane wave complex amplitudes and directions based on the sparse Bayesian framework using the variational Bayesian expectation and maximization method. Numerical simulations show that the proposed method can efficiently optimize the plane wave directions to reduce the basis mismatch and improve acoustic mode approximation accuracy. The proposed method involves slightly increased computational cost but obtains a higher reconstruction accuracy at extrapolated field points and is more robust under low signal-to-noise ratios compared with conventional methods. [ABSTRACT FROM AUTHOR]

The modeling of the capacitive phenomena, including the inductive effects becomes critical, especially in the case of a power converter with high switching frequencies, supplying an electrical device. At a low frequency, the electro-quasistatic (EQS) model is widely used to study the coupled resistive-capacitive effects, while the magneto-quasistatic (MQS) model is used to describe the coupled resistive-inductive effects. When the frequency increases, the Darwin model is preferred, which is able to capture the coupled resistive-capacitive-inductive effects by neglecting the radiation effects. In this work, we are interested in specifying the limits of these models, by investigating the influence of the frequency on the electromagnetic field distributions and the impedance of electromagnetic devices. Two different examples are carried out. For the first one, to validate the Darwin model, the measurement results are provided for comparison with the simulation results, which shows a good agreement. For the second one, the simulation results from three different models are compared, for both the local field distributions and the global impedances. It is shown that the EQS model can be used as an indicator to know at which frequency the Darwin model should be applied. [ABSTRACT FROM AUTHOR]

Coupling of electrical circuits with 2D and 3D computational domains is very important for practical applications. To this aim, the notions of "electrical current" and "voltage" must be defined precisely and linked with local quantities (i.e., fields and potentials) in the computational domain. Apart from the static case, the definition of voltage is more complex than it may appear at a first glance, and it is usually tainted by unspoken and/or not justified assumptions. The purpose of this work is twofold: on one hand, to shed light on the definition and on the physical meaning of voltage in the case of time varying quasi-static fields and, on the other hand, to show how to establish coupling equations between lumped parameters circuit model and 2D/3D computational domains. It is demonstrated that a precise physical significance can be given to the voltage in terms of power balance only (the notion of potential is unnecessary). A couple of original operators which allow to express voltages and currents are introduced. Based on a critical analysis of the research literature, it is shown that existing coupling formulas can all be rewritten as particular cases of these two operators. The developed analysis is independent from any computational method and can be used to devise new coupling formulas. [ABSTRACT FROM AUTHOR]

This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a systematic procedure which, starting from well-understood differential complexes such as the de Rham complex, derives new complexes and deduces the properties of the new complexes from the old. We relate the cohomology of the output complex to that of the input complexes and show that the new complex has closed ranges, and, consequently, satisfies a Hodge decomposition, Poincaré-type inequalities, well-posed Hodge–Laplacian boundary value problems, regular decomposition, and compactness properties on general Lipschitz domains. [ABSTRACT FROM AUTHOR]