Vector Finite Element Research Articles

A divergence-free vector finite-element method for efficient 3D magnetotelluric forward modeling

For large-scale magnetotelluric (MT) forward modeling using vector finite-element methods, iterative solvers are commonly applied. However, as frequency decreases close to zero, the iterative solution process struggles to converge. This is mainly caused by the fact that the weak conductivity term in the curl-curl equation governing the electromagnetic (EM) diffusion in the earth fades during the iterative solution for EM fields. This can lead to violation of the divergence-free condition for current and false jumps in the calculated fields. We develop a regularization technique for vector finite-element MT forward modeling, in which a scaled grad-div operator for electrical fields is included in the curl-curl equation to enforce the divergence-free condition explicitly. Because of its use of basis functions, the direct edge element discretization of the scaled term can lead to zero coefficients. To address this, an equivalent substitute of the scaled grad-div operator based on potential representation of electrical fields is used, discretized with node elements. For one specific element, this is finally reduced to a scaled gradient of the surface integral of current across all the surfaces of the element, resulting in better connectivity of the linear system of equations. The correctness of our algorithm is verified with two synthetic models and an inversion model from real data. The numerical performance is compared to the traditional iterative divergence correction technique (and without the application of the divergence correction for one case) for each of the three models. The results indicate that our algorithm is generally more efficient and stable compared to the traditional technique for all models at all periods considered, with significant improvement at long periods.

Efficient 3D Frequency Semi-Airborne Electromagnetic Modeling Based on Domain Decomposition

Landslides are common geological hazards that often result in significant casualties and economic losses. Due to their occurrence in complex terrain areas, conventional geophysical techniques face challenges in early detection and warning of landslides. Semi-airborne electromagnetic (SAEM) technology, utilizing unmanned aerial platforms for rapid unmanned remote sensing, can overcome the influence of complex terrain and serve as an effective approach for landslide detection and monitoring. In response to the low computational efficiency of conventional semi-airborne EM 3D forward modeling, this study proposes an efficient forward modeling method. To handle arbitrarily complex topography and geologic structures, the unstructured tetrahedron mesh is adopted to discretize the earth. Based on the vector finite element formula, the Dual–Primal Finite Element Tearing and Interconnecting (FETI-DP) method is further applied to enhance modeling efficiency. It obtains a reduced order subsystem and avoids directly solving huge overall linear equations by converting the entirety problem into the interface problem. We check our algorithm by comparing it with 1D semi-analytical solutions and the conventional finite element method. The numerical experiments reveal that the FETI-DP method utilities less memory and exhibits higher computation efficiency than the conventional methods. Additionally, we compare the electromagnetic responses with the topography model and flat earth model. The comparison results indicate that the effect of topography cannot be ignored. Further, we analyze the characteristic of electromagnetic responses when the thickness of the aquifer changes in a landslide area. We demonstrate the effectiveness of the 3D SAEM method for landslide detection and monitoring.

Design and Analysis of Complex End Region of Pumped Storage Generator Motor Based on Novel Electromagnetic Vector Method

A very large end leakage flux is produced by the end winding of a pumped storage generator motor. It has a significant impact on the design, flux density, and temperature in the end region of the pumped storage generator motor. The high temperature makes it difficult to design pumped storage generator motors. To quickly and accurately obtain the flux density of end components, a novel electromagnetic vector method is proposed in this paper. A 300 MW pumped storage generator motor is designed. The electromagnetic vector method and finite element method are used to calculate the flux density of the end components, respectively. The obtained results are compared and analyzed. The results reveal that the novel method is more convenient and faster than the finite element method. The novel electromagnetic vector method reduces the design time and calculation time of the end flux density. The flux density and losses of the end components in the pumped storage generator motor are studied under different operating conditions. Three-dimensional fluid and thermal coupled field is calculated and analyzed. Moreover, the calculation results are verified by the experimental values.

Pipe-soil interaction behaviors of deepwater J-lay pipeline on sloping seabed

The pipe-soil interaction behaviors are crucial factors for the J-lay pipeline design, which are mainly investigated on the horizontal seabed in previous studies. This paper develops an innovative method for the static and dynamic analyses of pipe-soil interactions considering the J-lay pipeline on a sloping seabed. This approach simulates the motion processes of the pipeline particles and solves the internal forces of the pipeline elements by the vector mechanics and finite element principle. The position coordinates formulations of pipeline particles are derived to calculate the pipeline locations on the sloping seabed. The non-linear hysteretic soil model and the modified Coulomb friction model are employed to simulate pipe-soil interactions in the vertical seabed plane as well as the axial and lateral directions. Moreover, a large-scale pipe-laying experimental test system is established to carry out the model experiments of a J-lay pipeline on different seabed conditions. The static and dynamic pipe-soil interaction responses are then analyzed by the VFIFE method for a 6-inch gas pipeline J-laying on the negative sloping seabed. The influence of seabed slope type and angle and soil characteristics is further explored on the pipe-soil interaction behaviors, which provide significant evidence for the design of pipe-lay on the sloping seabed.

Simulation of a Sensitive Mid-infrared (MIR) D-Shaped Optical Fiber Water Pollutant Sensor

In this work, an efficient optical sensor is proposed for the sensitive detection of various pollutants in water. The suggested optical sensor is based on an indium fluoride (InF3) glass fabricated as a D-shaped optical fiber. The polished surface of the D-shaped fiber is coated with a gold grating to induce the surface plasmon resonance (SPR). The SPR depends on the optical properties of the polluted water analyte in physical contact with the grating. The proposed optical SPR fiber sensor operates within the mid-infrared (MIR) range (3000–4500 nm) to detect any slight change in the water refractive index (RI) due to any pollutants. The full vectorial finite element method (FVFEM) is utilized to calculate the modal properties of the reported sensor. High sensor sensitivity of 17,834 nm/RIU (refractive index units) is achieved for the detection of dissolution of nitric acid (HNO3) in water at a concentration of 14% v/v (volume/volume). Additionally, the reported sensor detects the dissolution of hydrogen peroxide (H2O2) in water investigated at concentrations of 15% v/v and 30% v/v, with sensitivities of 12,308 nm/RIU and 17,143 nm/RIU, respectively. Further, suspending polystyrene beads of diameter 0.1 μm in the water at a concentration of 10% v/v gives a maximum sensitivity of 5333 nm/RIU. Therefore, the proposed sensor provides a promising approach for the detection of water pollutants in the MIR wavelength regime, rather than the weaker response in the near infrared.

Dark field imaging of high aspect ratio structures - a simple model.

A simplified model for dark-field optical imaging of three-dimensional high aspect ratio micro- and nano- structures is proposed, to reduce the time taken to simulate object fields with in-plane scattering between different parts of the object. Primary scattering is found by assuming that illumination of Manhattan geometries generates a set of spherical edge waves, following the incremental theory of diffraction. Secondary scattering is found by assuming that primary scattering is re-scattered from nearby features. Diffraction coefficients are simplified, and the number of illuminating beams is limited to those generating waves that enter the objective lens. Images obtained using TE and TM polarizations are compared, and results are benchmarked against a vectorial finite element model. Applications lie in simulating optical inspection of structures containing vertically etched features including MEMS and NEMS.

THz dual-core liquid photonic crystal fiber with high negative dispersion

A novel photonic crystal fiber (PCF) design is proposed and analyzed with highly negative dispersion for THz applications. The reported PCF has TOPAS background material due to its low material loss in THz regime. Further, dual porous cores are constructed and selectively infiltrated with liquid crystal (LC) material to control the dispersion characteristics of the reported PCF. The basic operation of the suggested dual core LCPCF (DC-LCPCF) depends on the optical coupling between the supported modes of the two porous core regions in the THz regime to achieve high negative dispersion for the two fundamental polarizations: transverse electric (TE) and transverse magnetic (TM). The coupling can be switched between the TE and TM modes by applying an external electric field on the LC material via two metallic electrodes. The full vectorial finite element method (FVFEM) is utilized to study the dispersion characteristics of the DC-LCPCF structure. The obtained results reveal that the TE and TM modes have large negative chromatic dispersions of − 44.57 ps/THz/cm and − 30.59 ps/THz/cm at frequencies of 0.386 THz and 0.4027 THz, respectively. So, it will be a solution for further innovation of fiber devices in the THz regime.

Approximation properties of primal discontinuous Petrov-Galerkin method on general quadrilateral meshes

We consider the approximation properties of primal discontinuous Petrov-Galerkin (DPG) method on quadrilateral meshes. We show how the previous convergence results as well as the Aubin-Nitsche type duality arguments can be extended to cover arbitrary convex quadrilateral elements with bilinear isomorphisms. The arguments are based on the approximation theory of quadrilateral vector finite element spaces associated to the numerical flux variable of the DPG approximation. The theoretical results are validated by a numerical experiment that features also a comparison between the primal DPG method and a conventional least squares finite element method with the same number of degrees of freedom.

Three Low Order H-Curl-Curl Finite Elements on Triangular Meshes

Abstract We construct three H-curl-curl finite elements. The P 2 P_{2} and P 3 P_{3} vector finite element spaces are both enriched by one common P 4 P_{4} bubble and their local degrees of freedom are 13 and 21, respectively. As there does not exist any P 1 P_{1} H-curl-curl conforming finite element, the P 1 P_{1} H-curl-curl nonconforming finite element is constructed with three additional P 4 P_{4} bubbles. Numerical tests are presented, confirming the conformity and the optimal order of convergence.

Low-loss and high-sensitivity surface plasmon resonance sensor composed of microstructured optical fibers for wide visible-to-infrared and refractive index ranges

ObjectiveOptical fiber sensing technology develops rapidly and is widely used. In order to design a SPR sensor with better performance and easy production, a new type D-MOF sensor with low loss, high sensitivity and simple structure is proposed. MethodsThe full vector finite element method (FEM) was used to optimize the structure parameters by using the multi-physics analysis software COMSOL, and the performance of the structure was systematically analyzed. ResultsThe sensor operational wavelength spans the visible and infrared range (430 nm to 7200 nm). The refractive index detection range of 1.00–1.42, the maximum wavelength sensitivity and the optimal resolution are 60,000 nm/RIU and 1.67 × 10−6 RIU−1, respectively, and the maximum limiting loss peak is 64.99 dB/cm. In addition, MOF-SPR sensors have large tolerance and low sensitivity of structural parameters in actual manufacturing. It has great application potential in biosensing and other fields.

Mid-infrared water pollutant sensor based on SPR-PCF

In this paper, a highly sensitive water pollutant optical sensor is proposed and analyzed. The suggested sensor consists of photonic crystal fiber with a core surrounded by four elliptical holes infiltrated with the studied analyte (pure/polluted water sample). In addition, two gold nanorods are mounted horizontally at the inner surfaces of two horizontal elliptical holes. The proposed sensor can efficiently detect dissolved pollutants in water such as nitric acid ({mathrm{HNO}}_{3}) with concentrations of 14, 23 and 35% and {mathrm{H}}_{2}{mathrm{O}}_{2} with concentrations of 7, 15 and 30%. The dissolved pollutants in water (analyte) have refractive indices (RIs) in the range of 1.350–1.355 in the mid infrared regime from λ = 2200 nm to λ = 3500 nm. A fully vectorial finite element method (FVFEM) is employed for the modal analysis of the reported structure. The geometrical parameters are studied to maximize the RI sensitivity where a high sensitivity of 36,000 nm/RIU is achieved between the studied RIs of 1.350 and 1.355. The obtained RI sensitivity is higher than those of the recent reported sensors in the literature especially those operating in the mid infrared wavelengths.

3D modeling of vector/edge finite element method for multi-ablation technique for large tumor-computational approach.

Microwave ablation (MWA) is a cancer thermal ablation treatment that uses electromagnetic waves to generate heat within the tissue. The goal of this treatment is to eliminate tumor cells while leaving healthy cells unharmed. During MWA, excess heat generation can kill healthy cells. Hence, mathematical models and numerical techniques are required to analyze the heat distribution in the tissue before the treatment. The aim of this research is to explain the implementation of the 3D vector finite element method in a wave propagation model that simulates the specific absorption rate in the liver. The 3D Nedelec elements from H(curl; Ω) space are used to discretize the wave propagation model, and this implementation is helpful in solving many real-world problems that involve electromagnetic propagation with perfect conducting and absorbing boundary conditions. One of the difficulties in ablation treatment is creating a large ablation zone for a large tumor (diameter greater than 3 cm) in a short period of time with minimum damage to the surrounding tissue. This article addresses the aforementioned issue by introducing four antennas into the different places of the tumor sequentially and producing heat uniformly over the tumor. The results demonstrated that 95.5% of the tumor cells were killed with minimal damage to the healthy cells when the heating time was increased to 4 minutes at each position. Subsequently, we studied the temperature distribution and localised tissue contraction in the tissue using the three-dimensional bio-heat equation and temperature-time dependent model, respectively. The local tissue contraction is measured at arbitrary points in the domain and is more noticeable at temperatures higher than 102°C. The thermal damage in the liver during MWA treatment is investigated using the three-state cell death model. The system of partial differential equations is solved numerically due to the complex geometry of the domain, and the results are compared with experimental data to validate the models and parameters.

High Sensitivity and Wide Range Refractive Index Sensor Based on Surface Plasmon Resonance Photonic Crystal Fiber.

In order to detect the refractive index (RI) of high refractive index materials such as trichlorobenzene and aniline in the near-infrared and mid-infrared spectra and expand the detection range of the refractive index, a surface plasmon resonance (SPR) photonic crystal fiber (PCF) sensor based on an elliptical sensing channel is proposed for high refractive index detection. The fiber core and the analyte channel are surrounded by two types of air holes with different sizes. When the surface plasmon resonance effect appears at the interface between the fiber core and the elliptical sensing layer, obvious resonance peaks appear in the near-infrared and mid-infrared bands. The full vector finite element method (FEM) is used to study the sensing characteristics of the sensor and the influence of structural parameters on the resonance peak. The results demonstrate that the sensor achieves detection in the refractive index range of 1.41-1.58, in the wavelength range of 1600-3200 nm. The average wavelength sensitivity is 9217.22 nm/RIU, and the refractive index resolution is 10.85 × 10-6 RIU. The proposed sensor realizes high refractive index detection in the near-infrared and mid-infrared bands, and obtains an ultra-wide detection range and higher sensitivity. The sensor has broad application prospects in chemical detection, biomedical sensing and other fields, and provides a theoretical reference for the design of a photonic crystal fiber surface plasmon resonance sensor.

Design of an octagon structure dual-core photonic crystal fiber polarization beam splitter with a liquid crystal filled central hole

In this paper, an octagon structure dual-core photonic crystal fiber (OS-DC-PCF) polarization beam splitter (PBS) with a liquid crystal filled central hole is designed. The octagon lattice structure is simple and easy to fabricate. By filling the nematic liquid crystal into the central hole, the birefringence of the octagon structure can be enhanced, and the coupling lengths along the two polarization directions can be reduced. The full vector-finite element method is used to analyze the polarization and coupling characteristics, and the influences of different structural parameters on the extinction ratio (ER) of the PBS. The simulation results show that for the designed OS-DC-PCF PBS, the maximum ER is ∼91 dB, the splitting length is only 125μm, and the working bandwidth is up to 339 nm (1.493–1.832μm), covering the whole C+L+U communication bands. Moreover, it has good fabrication tolerances and anti-interference ability. It is believed that the designed OS-DC-PCF PBS can provide the critical device support for the future all-optical communication systems.

Advanced Modeling of SIW Structures by 2-D Finite Element Method

This article presents an advanced modeling of substrate-integrated waveguide (SIW) structures by the 2-D vector finite element method (FEM). All types of losses are considered, together with input/output coupling by the rectangular waveguide and coaxial cable. Multiple dielectric regions can be defined, allowing the modeling of partially air-filled SIW structures with excellent accuracy and efficiency.

Terahertz photonic crystal fiber for sensing the creatinine level in the blood

In this work, THz photonic crystal fiber (PCF) is used to detect the creatinine level in the blood with high sensitivity. The sensing technique depends on increasing the light interaction with the analyte infiltrated into the air holes in the fiber core region. In this regard, most of the light power should be confined through the analyte region. This will increase the relative sensitivity coefficient that is proportional to the analyte power fraction. The operation of the suggested sensor is based on studying the light–analyte interaction at different creatinine concentrations by detecting the change of the analyte power fraction against the change of the creatinine level in the blood at the THz range 0.5–1.5 THz. The effective mode index (neff), effective material loss (EML), effective modal area (Aeff), and relative sensor sensitivity are calculated using the full vectorial finite element method. Additionally, the different geometrical parameters are studied to maximize the sensor’s sensitivity. The proposed THz-PCF has a 93% and 95% relative sensitivity for x- and y-polarized modes, respectively. Therefore, the suggested THz-PCF biosensor gives a promising usage in measuring the creatinine level in the blood.

Optical mid-infrared modulator based on D-shaped photonic crystal fiber and GST phase changing material

Photonic crystal fibers (PCFs) have recently attracted compelling attention because of their numerous applications, particularly in the mid-infrared (mid-IR) wavelength region. In this paper, we have presented and analyzed mid-IR optical modulator based on phase-changing material (PCM) known as germanium-antimony-tellurium (GST) and D-shaped PCF. The modulation process can be performed as the GST material’s phase undergoes a transition between amorphous (on) and crystalline (off) states. To analyze the proposed design numerically, full vectorial finite element method (FVFEM) is employed. Further, we studied the light propagation through the suggested structure using 3D finite difference time domain (FDTD) method. The optical losses of the fundamental transverse electric (TE) mode supported by the reported structure in the two GST states are studied. The obtained extinction ratio (ER) of the proposed modulator approaches 302.61 dB, whereas the insertion loss (IL) is less than 0.00014 dB throughout the wavelength range from 3 to 5.8 μm at a device length (LD) of 0.2 mm. Therefore, the suggested modulator can be utilized in photonic integrated circuits that require high ER, very low IL, and large optical bandwidth.

Double-clad photonic crystal fibre for two-photon microscopy

Miniaturized and head-mountable scanning nonlinear fibre-optic microscopy offers an exciting opportunity for the visualization of brain activity on freely walking mice. One challenge in the creation of such miniaturized microscopes is the design of optical fibres capable of delivering ultrashort pulses without distortion and collecting fluorescence simultaneously and efficiently. A double-clad photonic crystal fibre (DCPCF) with 37-cell air holes arranged in hexagonal lattice in core is proposed. The full vector finite element method (FEM) is used for numerical analysis. Nearly zero dispersion of −0.34 ps/nm/km at 920 nm wavelength is finally achieved. The single-mode core with an NA of 0.53 at 920 nm wavelength is used to deliver the femtosecond excitation beam and the inner cladding with an NA of 0.54 at 532 nm is used to collect the two-photon fluorescence signal. The nearly zero dispersion, high NA and low nonlinearity will be beneficial to promote its application in two-photon microscopy.

End-to-end GPU acceleration of low-order-refined preconditioning for high-order finite element discretizations

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.

Controlled-source electromagnetic modeling using a versatile secondary electric-field formulation and efficient multigrid-based preconditioner

Frequency-domain controlled-source electromagnetic (CSEM) surveying is a widely used geophysical electromagnetic exploration method. To avoid the singularity near the transmitting sources and improve the accuracy of three-dimensional (3D) CSEM forward modeling, we adopt the vector finite element method and secondary electric field formulation combined with hexahedral grids to solve the frequency-domain Maxwell equations. Primary field computation applies the fast Hankel transform using digital linear filters to quickly calculate the electromagnetic field generated by different electromagnetic dipoles or loops in the layered medium. Therefore, our forward modeling algorithm is highly versatile and compatible with various electromagnetic sources and scenarios of CSEM application (e.g., airborne, offshore, or marine). To improve the convergence rate, we use an algebraic multigrid solver based on the established Hiptmair–Xu decomposition. Using direct solvers requires a vast amount of memory, while conventional iterative solvers usually fail or converge only very slowly. Our new iterative scheme does not require the divergence corrections commonly applied in finite difference or finite volume electromagnetic modeling to accelerate the convergence. We implement complete parallelism in our finite element programming to efficiently solve the CSEM forward problems on massively parallel clusters. We use different numerical methods to obtain high-precision comparison results for one-dimensional and typical 3D models to demonstrate our algorithm's effectiveness and correctness. Finally, a public model of 3 million cells is used to simulate the responses of marine CSEM surveying. The proposed new solver shows robust scalability when using nearly 1000 cores on the cluster, demonstrating that our algorithm is particularly suitable for large-scale 3D forward modeling.