Accuracy of dipole source reconstruction in the 3-layer BEM model against the 5-layer BEM-FMM model.

Objective: To compare cortical dipole fitting spatial accuracy between the widely used yet highly simplified 3-layer and modern more realistic 5-layer BEM-FMM models with and without adaptive mesh refinement (AMR) methods. Methods: We generate simulated noiseless 256-channel EEG data from 5-layer (7-compartment) meshes of 15 subjects from the Connectome Young Adult dataset. For each subject, we test four dipole positions, three sets of conductivity values, and two types of head segmentation. We use the boundary element method (BEM) with fast multipole method (FMM) acceleration, with or without (AMR), for forward modeling. Dipole fitting is carried out with the FieldTrip MATLAB toolbox. Results: The average position error (across all tested dipoles, subjects, and models) is ~4 mm, with a standard deviation of ~2 mm. The orientation error is ~20° on average, with a standard deviation of ~15°. Without AMR, the numerical inaccuracies produce a larger disagreement between the 3- and 5-layer models, with an average position error of ~8 mm (6 mm standard deviation), and an orientation error of 28° (28° standard deviation). Conclusions: The low-resolution 3-layer models provide excellent accuracy in dipole localization. On the other hand, dipole orientation is retrieved less accurately. Therefore, certain applications may require more realistic models for practical source reconstruction. AMR is a critical component for improving the accuracy of forward EEG computations using a high-resolution 5-layer volume conduction model. Significance: Improving EEG source reconstruction accuracy is important for several clinical applications, including epilepsy and other seizure-inducing conditions.

A hybrid boundary element-finite element approach for solving the EEG forward problem in brain modeling.

This article introduces a hybrid BE-FE method for solving the EEG forward problem, leveraging the strengths of both the Boundary Element Method (BEM) and Finite Element Method (FEM). FEM accurately models complex and anisotropic tissue properties for realistic head geometries, while BEM excels in handling isotropic tissue regions and dipolar sources efficiently. The proposed hybrid method divides regions into homogeneous boundary element (BE) regions that include sources and heterogeneous anisotropic finite element (FE) regions. So, BEM models the brain, including dipole sources, and FEM models other head layers. Validation includes inhomogeneous isotropic/anisotropic three- and four-layer spherical head models, and a four-layer MRI-based realistic head model. Results for six dipole eccentricities and two orientations are computed using BEM, FEM, and hybrid BE-FE method. Statistical analysis, comparing error criteria of RDM and MAG, reveals notable improvements using the hybrid FE-BE method. In the spherical head model, the hybrid BE-FE method compared with FEM demonstrates enhancements of at least 1.05 and 38.31% in RDM and MAG criteria, respectively. Notably, in the anisotropic four-layer head model, improvements reach a maximum of 88.3% for RDM and 93.27% for MAG over FEM. Moreover, in the anisotropic four-layer realistic head model, the proposed hybrid method exhibits 55.4% improvement in RDM and 89.3% improvement in MAG compared to FEM. These findings underscore the proposed method is a promising approach for solving the realistic EEG forward problems, advancing neuroimaging techniques and enhancing understanding of brain function.

Multi-domain wavelet boundary element method for calculating two-dimensional stress intensity factors.

In order to improve the accuracy of stress intensity factors (SIFs) calculated by traditional boundary element methods (BEM), the multi-domain wavelet boundary element method (WBEM) is proposed. Firstly, by adjusting the nodes of the B-spline wavelet element on the interval, crack-tip elements are constructed. Since B-spline wavelet on the interval (BSWI) has excellent compact support characteristics and is particularly suitable for describing solution domains with large gradient changes, the constructed crack-tip can reduce the numerical oscillation effect near the crack tip. Secondly, the crack-tip elements are implemented into WBEM. And the combination of WBEM and multi-domain technology can effectively handle interface cracks. Thirdly, the crack problem solving strategy based on multi-domain WBEM can directly evaluate the SIFs of cracks. Finally, several numerical examples involving homogeneous media and bi-material models are given to verify that the proposed method is simple and highly accurate.

An adaptive h-refinement method for the boundary element fast multipole method for quasi-static electromagnetic modeling.

Objective.In our recent work pertinent to modeling of brain stimulation and neurophysiological recordings, substantial modeling errors in the computed electric field and potential have sometimes been observed for standard multi-compartment head models. The goal of this study is to quantify those errors and, further, eliminate them through an adaptive mesh refinement (AMR) algorithm. The study concentrates on transcranial magnetic stimulation (TMS), transcranial electrical stimulation (TES), and electroencephalography (EEG) forward problems.Approach.We propose, describe, and systematically investigate an AMR method using the boundary element method with fast multipole acceleration (BEM-FMM) as the base numerical solver. The goal is to efficiently allocate additional unknowns to critical areas of the model, where they will best improve solution accuracy. The implemented AMR method's accuracy improvement is measured on head models constructed from 16 Human Connectome Project subjects under problem classes of TES, TMS, and EEG. Errors are computed between three solutions: an initial non-adaptive solution, a solution found after applying AMR with a conservative refinement rate, and a 'silver-standard' solution found by subsequent 4:1 global refinement of the adaptively-refined model.Main results.Excellent agreement is shown between the adaptively-refined and silver-standard solutions for standard head models. AMR is found to be vital for accurate modeling of TES and EEG forward problems for standard models: an increase of less than 25% (on average) in number of mesh elements for these problems, efficiently allocated by AMR, exposes electric field/potential errors exceeding 60% (on average) in the solution for the unrefined models.Significance.This error has especially important implications for TES dosing prediction-where the stimulation strength plays a central role-and for EEG lead fields. Though the specific form of the AMR method described here is implemented for the BEM-FMM, we expect that AMR is applicable and even required for accurate electromagnetic simulations by other numerical modeling packages as well.

Development of a Helmet-Shape Dual-Channel RF coil for brain imaging at 54 mT using inverse boundary element method.

Very-low field (VLF) magnetic resonance imaging (MRI) offers advantages in term of size, weight, cost, and the absence of robust shielding requirements. However, it encounters challenges in maintaining a high signal-to-noise ratio (SNR) due to low magnetic fields (below 100 mT). Developing a close-fitting radio frequency (RF) receive coil is crucial to improve the SNR. In this study, we devised and optimized a helmet-shaped dual-channel RF receive coil tailored for brain imaging at a magnetic field strength of 54 mT (2.32 MHz). The methodology integrates the inverse boundary element method (IBEM) to formulate initial coil structures and wiring patterns, followed by optimization through introducing regularization terms. This approach frames the design process as an inverse problem, ensuring a close fit to the head contour. Combining theoretical optimization with physical measurements of the coil's AC resistance, we identified the optimal loop count for both axial and radial coils as nine and eight loops, respectively. The effectiveness of the designed dual-channel coil was verified through the imaging of a CuSO4 phantom and a healthy volunteer's brain. Notably, the in-vivo images exhibited an approximate 16-25 % increase in SNR with poorer B1 homogeneity compared to those obtained using single-channel coils. The high-quality images achieved by T1, T2-weighted, and fluid-attenuated inversion-recovery (FLAIR) protocols enhance the diagnostic potential of VLF MRI, particularly in cases of cerebral stroke and trauma patients. This study underscores the adaptability of the design methodology for the customization of RF coil structures in alignment with individual imaging requirements.

Vial Wall Effect on Freeze-Drying Speed.

The vial wall thermal conductivity and thickness effect on freeze-drying speed is simulated. A 2D axisymmetric numerical simulation of Mannitol freeze-drying is employed using the boundary element method. The originality of the presented approach lies in the simulation of heat transfer in the vial walls as an additional computational domain in contrast to the typical methodology without a vial wall. The numerical model was validated using our measurements and the measurements from the literature. Increasing the glass vial thickness from 1 mm to 2 mm has been found as the major factor in primary drying time, increasing the gravimetrical Kv up to 20 % for all the simulated chamber pressures. The effect of thermal conductivity was simulated using a polymer and aluminium vial replacing the standard glass vial of the same thickness. The polymer vial's decreased Kv value is 5.6 % at a low chamber pressure of 50 mTorr, and 12.2 % at 400 mTorr, which is in excellent agreement with the experiment. Using higher conductivity materials, for example, aluminium, only 3.7 % and 2.3 % Kv increase were computed for low and high chamber pressures respectively.

Coupling finite and boundary element methods to solve the Poisson-Boltzmann equation for electrostatics in molecular solvation.

The Poisson-Boltzmann equation is widely used to model electrostatics in molecular systems. Available software packages solve it using finite difference, finite element, and boundary element methods, where the latter is attractive due to the accurate representation of the molecular surface and partial charges, and exact enforcement of the boundary conditions at infinity. However, the boundary element method is limited to linear equations and piecewise constant variations of the material properties. In this work, we present a scheme that couples finite and boundary elements for the linearised Poisson-Boltzmann equation, where the finite element method is applied in a confined solute region and the boundary element method in the external solvent region. As a proof-of-concept exercise, we use the simplest methods available: Johnson-Nédélec coupling with mass matrix and diagonal preconditioning, implemented using the Bempp-cl and FEniCSx libraries via their Python interfaces. We showcase our implementation by computing the polar component of the solvation free energy of a set of molecules using a constant and a Gaussian-varying permittivity. As validation, we compare against well-established finite difference solvers for an extensive binding energy data set, and with the finite difference code APBS (to 0.5%) for Gaussian permittivities. We also show scaling results from protein G B1 (955 atoms) up to immunoglobulin G (20,148 atoms). For small problems, the coupled method was efficient, outperforming a purely boundary integral approach. For Gaussian-varying permittivities, which are beyond the applicability of boundary elements alone, we were able to run medium to large-sized problems on a single workstation. The development of better preconditioning techniques and the use of distributed memory parallelism for larger systems remains an area for future work. We hope this work will serve as inspiration for future developments that consider space-varying field parameters, and mixed linear-nonlinear schemes for molecular electrostatics with implicit solvent models.

Numerical Solution of the Electric Field and Dielectrophoresis Force of Electrostatic Traveling Wave System.

Electrostatic traveling wave (ETW) methods have shown promising performance in dust mitigation of solar panels, particle transport and separation in in situ space resource utilization, cell manipulation, and separation in biology. The ETW field distribution is required to analyze the forces applied to particles and to evaluate ETW design parameters. This study presents the numerical results of the ETW field distribution generated by a parallel electrode array using both the charge simulation method (CSM) and the boundary element method (BEM). A low accumulated error of the CSM is achieved by properly arranging the positions and numbers of contour points and fictitious charges. The BEM can avoid the inconvenience of the charge position required in the CSM. The numerical results show extremely close agreement between the CSM and BEM. For simplification, the method of images is introduced in the implementation of the CSM and BEM. Moreover, analytical formulas are obtained for the integral of Green's function along boundary elements. For further validation, the results are cross-checked using the finite element method (FEM). It is found that discrepancies occur at the ends of the electrode array. Finally, analyses are provided of the electric field and dielectrophoretic (DEP) components. Emphasis is given to the regions close to the electrode surfaces. These results provide guidance for the fabrication of ETW systems for various applications.

Towards Digital Twins of Plasmonic Sensors: Constructing the Complex Numerical Model of a Plasmonic Sensor Based on Hexagonally Arranged Gold Nanoparticles.

In this work, we aim to design the digital twin of a plasmonic sensor based on hexagonally arranged ellipsoidal gold nanoparticles fixed to a glass substrate. Based on electron microscopy images of three experimentally realized nanoparticle arrangement types, we constructed numerical models in environments based on finite element method (FEM) and boundary element method (BEM), namely COMSOL Multiphysics for FEM and the MNPBEM Matlab Toolbox for BEM. Models with nonperiodic and periodic boundary conditions with different unit cells were constructed to study the plasmonic behavior of both the single ellipsoidal particles and the hexagonal nanoparticle arrangements. The effect of the geometrical parameters, namely the interparticle distance, the nanoparticle diameter and thickness, on the resulting LSPR peak positions and bulk refractive index sensitivities were studied in detail, also taking into account the effect of the SiO2 substrate (pillars) under the ellipsoidal particles. We have demonstrated that by optimizing the models, the LSPR peak positions (and sensitivities) can match the experimentally measured values within 1 nm (nm/RIU) precision. The comparison of simulation conditions and the detailed discussion of the effect of the geometrical parameters and used gold dielectric functions on the obtained sensitivities can be very beneficial for the optimization of plasmonic sensor constructions through numerical simulations.

On the noise generation and unsteady performance of combined heaving and pitching foils.

A transient two-dimensional acoustic boundary element solver is coupled to a potential flow boundary element solver via Powell's acoustic analogy to determine the acoustic emission of isolated hydrofoils performing biologically-inspired motions. The flow-acoustic boundary element framework is validated against experimental and asymptotic solutions for the noise produced by canonical vortex-body interactions. The numerical framework then characterizes the noise production of an oscillating foil, which is a simple representation of a fish caudal fin. A rigid NACA 0012 hydrofoil is subjected to combined heaving and pitching motions for Strouhal numbers (0.03

Effects of head modeling errors on the spatial frequency representation of MEG.

Objectives.We aim to investigate the effects of head model inaccuracies on signal and source reconstruction accuracies for various sensor array distances to the head. This allows for the assessment of the importance of head modeling for next-generation magnetoencephalography (MEG) sensors, optically-pumped magnetometers (OPM).Approach.A 1-shell boundary element method (BEM) spherical head model with 642 vertices of radius 9 cm and conductivity of 0.33 S m-1was defined. The vertices were then randomly perturbed radially up to 2%, 4%, 6%, 8% and 10% of the radius. For each head perturbation case, the forward signal was calculated for dipolar sources located at 2 cm, 4 cm, 6 cm and 8 cm from the origin (center of the sphere), and for a 324 sensor array located at 10 cm to 15 cm from the origin. Equivalent current dipole (ECD) source localization was performed for each of these forward signals. The signal for each perturbed spherical head case was then analyzed in the spatial frequency domain, and the signal and ECD errors were quantified relative to the unperturbed case.Main results.In the noiseless and high signal-to-noise ratio (SNR) case of approximately ≥6 dB, inaccuracies in our spherical BEM head conductor models lead to increased signal and ECD inaccuracies when sensor arrays are placed closer to the head. This is true especially in the case of deep and superficial sources. In the noisy case however, the higher SNR for closer sensor arrays allows for an improved ECD fit and outweighs the effects of head geometry inaccuracies.Significance.OPMs may be placed directly on the head, as opposed to the more commonly used superconducting quantum interference device sensors which must be placed a few centimeters away from the head. OPMs thus allow for signals of higher spatial resolution to be captured, resulting in potentially more accurate source localizations. Our results suggest that an increased emphasis on accurate head modeling for OPMs may be necessary to fully realize its improved source localization potential.

Asymmetrical Plasmon Distribution in Hybrid AuAg Hollow/Solid Coded Nanotubes.

Morphological control at the nanoscale paves the way to fabricate nanostructures with desired plasmonic properties. In this study, we discuss the nanoengineering of plasmon resonances in 1D hollow nanostructures of two different AuAg nanotubes, including completely hollow nanotubes and hybrid nanotubes with solid Ag and hollow AuAg segments. Spatially resolved plasmon mapping by electron energy loss spectroscopy (EELS) revealed the presence of high order resonator-like modes and localized surface plasmon resonance (LSPR) modes in both nanotubes. The experimental findings accurately correlated with the boundary element method (BEM) simulations. Both experiments and simulations revealed that the plasmon resonances are intensely present inside the nanotubes due to plasmon hybridization. Based on the experimental and simulated results, we show that the novel hybrid AuAg nanotubes possess two significant coexisting features: (i) LSPRs are distinctively generated from the hollow and solid parts of the hybrid AuAg nanotube, which creates a way to control a broad range of plasmon resonances with one single nanostructure, and (ii) the periodicity of the high-order modes are disrupted due to the plasmon hybridization by the interaction of solid and hollow parts, resulting in an asymmetrical plasmon distribution in 1D nanostructures. The asymmetry could be modulated/engineered to control the coded plasmonic nanotubes.

Streamlines of the Poynting Vector and Chirality Flux around a Plasmonic Bowtie Nanoantenna.

The streamlines of the energy flux (Poynting vectors) and chirality flux as well as the intensity of the electric field around various plasmonic nanostructures (nanocube, nanocuboid, nanotriangle, hexagonal nanoplate and bowtie nanoantenna) induced by a circularly polarized (CP) or linearly polarized (LP) light were studied theoretically. The boundary element method combined with the method of moment was used to solve a set of surface integral equations, based on the Stratton-Chu formulation, for analyzing the highly distorted electromagnetic (EM) field in the proximity of these nanostructures. We discovered that the winding behavior of these streamlines exhibits versatility for various modes of the surface plasmon resonance of different nanostructures. Recently, using plasmonic nanostructures to facilitate a photochemical reaction has gained significant attention, where the hot carriers (electrons) play important roles. Our findings reveal a connection between the flow pattern of energy flux and the morphology of the photochemical deposition around various plasmonic nanostructures irradiated by a CP light. For example, numerical results exhibit vertically helical streamlines of the Poynting vector around an Au nanocube and transversely twisted-roll streamlines around a nanocuboid. Additionally, the behaviors of the winding energy and chirality fluxes at the gap and corners of a plasmonic bowtie nanoantenna, implying a highly twisted EM field, depend on the polarization of the incident LP light. Our analysis of the streamlines of the Poynting vector and chirality flux offers an insight into the formation of plasmon-enhanced photocatalysis.

Efficient Model Assisted Probability of Detection Estimations in Eddy Current NDT with ACA-SVD Based Forward Solver.

Model assisted probability of detection (MAPoD) is crucial for quantifying the inspection capability of a nondestructive testing (NDT) system which uses the coil or probe to sense the size and location of the cracks. Unfortunately, it may be computationally intensive for the simulation models. To improve the efficiency of the MAPoD, in this article, an efficient 3D eddy current nondestructive evaluation (ECNDE) forward solver is proposed to make estimations for PoD study. It is the first time that singular value decomposition (SVD) is used as the recompression technique to improve the overall performance of the adaptive cross approximation (ACA) algorithm-based boundary element method (BEM) ECNDE forward solver for implementation of PoD. Both the robustness and efficiency of the proposed solver are demonstrated and testified by comparing the predicted impedance variations of the coil with analytical, semi-analytical and experimental benchmarks. Calculation of PoD curves assisted by the proposed simulation model is performed on a finite thickness plate with a rectangular surface flaw. The features, which are the maximum impedance variations of the coil for various flaw lengths, are obtained entirely by the proposed model with selection of the liftoff distance as the uncertain parameter in a Gaussian distribution. The results show that the proposed ACA-SVD based BEM fast ECNDE forward solver is an excellent simulation model to make estimations for MAPoD study.

Three-Dimensional Boundary Element Strategy for Stress Sensitivity of Fractional-Order Thermo-Elastoplastic Ultrasonic Wave Propagation Problems of Anisotropic Fiber-Reinforced Polymer Composite Material.

A new three-dimensional (3D) boundary element method (BEM) strategy was developed to solve fractional-order thermo-elastoplastic ultrasonic wave propagation problems based on the meshless moving least squares (MLS) method. The temperature problem domain was divided into a number of circular sub-domains. Each node was the center of the circular sub-domain surrounding it. The Laplace transform method was used to solve the temperature problem. A unit test function was used in the local weak-form formulation to generate the local boundary integral equations, and the inverse Laplace transformation method was used to find the transient temperature solutions. Then, the three-dimensional elastoplastic problems could be solved using the boundary element method (BEM). Initial stress and strain formulations are adopted, and their distributions are interpolated using boundary integral equations. The effects of the fractional-order parameter and anisotropy are investigated. The proposed method's validity and performance are demonstrated for a two-dimensional problem with excellent agreement with other experimental and numerical results.

Numerical calculation of near-field in vicinity of three-dimensional nano-optical probes using boundary element method.

The diffraction limit is one of the main obstacles in the development of microscopes to analyze the morphology and structure of materials. The main idea of near field scanning optical microscopy (NSOM) is to overcome the diffraction limit using sub-wavelength apertures. In this work, the near-field is simulated in the vicinity of three-dimensional nano-optical apertureless probes. For this purpose, the Helmholtz equation is solved using the boundary element method (BEM). The effects of different parameters on the near field generated in the vicinity of the optical probe are studied. These parameters consist of the length and radius of the probe, the size of the aperture, the angle of the tapered tip, and the geometry of the probe tip. The main advantages of the proposed method are the high accuracy, the very short calculation time, and the ability to calculate the near field inside and outside the optical probe without any approximation.

CoilGen: Open-source MR coil layout generator.

PURPOSE: An automated algorithm for generating realizable MR gradient and shim coil layouts based on the boundary element method is presented here. The overall goal is to reduce postprocessing effort and thus enable for rapid prototyping of new coil designs. For a given surface mesh and target field, the algorithm generates a connected, non-overlapping wire path. METHODS: The proposed algorithm consists of several steps: Stream function optimization, two-dimensional surface projection, potential discretization, topological contour sorting, opening and interconnecting contours, and finally adding non-overlapping return paths. Several technical parameters such as current strength, inductance and field accuracy are assessed for quality control. RESULTS: The proposed method is successfully demonstrated in four different examples. All exemplary results demonstrate high accuracy with regard to reaching the respective target field. The optimal discretization for a given stream function is found by generating multiple layouts while varying the input parameter values. CONCLUSION: The presented algorithm allows for a rapid generation of interconnected coil layouts with high flexibility and low discretization error. This enables to reduce the overall post-processing effort. The source code of this work is publicly available ( https://github.com/Philipp-MR/CoilGen).

Simulation Study of Different OPM-MEG Measurement Components.

Magnetoencephalography (MEG) is a neuroimaging technique that measures the magnetic fields of the brain outside of the head. In the past, the most suitable magnetometer for MEG was the superconducting quantum interference device (SQUID), but in recent years, a new type has also been used, the optically pumped magnetometer (OPM). OPMs can be configured to measure multiple directions of magnetic field simultaneously. This work explored whether combining multiple directions of the magnetic field lowers the source localization error of brain sources under various conditions of noise. We simulated dipolar-like sources for multiple configurations of both SQUID- and OPM-MEG systems. To test the performance of a given layout, we calculated the average signal-to-noise ratio and the root mean square of the simulated magnetic field; furthermore, we evaluated the performance of the dipole fit. The results showed that the field direction normal to the scalp yields a higher signal-to-noise ratio and that ambient noise has a much lower impact on its localization error; therefore, this is the optimal choice for source localization when only one direction of magnetic field can be measured. For a low number of OPMs, combining multiple field directions greatly improves the source localization results. Lastly, we showed that MEG sensors that can be placed closer to the brain are more suitable for localizing deeper sources.

Boundary Element and Sensitivity Analysis of Anisotropic Thermoelastic Metal and Alloy Discs with Holes.

The main aim of this paper was to develop an advanced processing method for analyzing of anisotropic thermoelastic metal and alloy discs with holes. In the boundary element method (BEM), the heat impact is expressed as an additional volume integral in the corresponding boundary integral equation. Any attempt to integrate it directly will necessitate domain discretization, which will eliminate the BEM's most distinguishing feature of boundary discretization. This additional volume integral can be transformed into the boundary by using branch-cut redefinitions to avoid the use of additional line integrals. The numerical results obtained are presented graphically to show the effects of the transient and steady-state heat conduction on the quasi-static thermal stresses of isotropic, orthotropic, and anisotropic metal and alloy discs with holes. The validity of the proposed technique is examined for one-dimensional sensitivity, and excellent agreement with finite element method and experimental results is obtained.

DC-electrokinetic motion of colloidal cylinder(s) in the vicinity of a conducting wall.

The boundary effects on DC-electrokinetic behavior of colloidal cylinder(s) in the vicinity of a conducting wall is investigated through a computational model. The contribution of the hydrodynamic drag, gravity, electrokinetic (i.e., electrophoretic and dielectrophoretic), and colloidal forces (i.e., forces due to the electrical double layer and van der Waals interactions) are incorporated in the model. The contribution of electrokinetic and colloidal forces are included by introducing the resulting forces as an external force acting on the particle(s). The colloidal forces are implemented with the prescribed expressions from the literature, and the electrokinetic force is obtained by integrating the corresponding Maxwell stress tensor over the particles' surfaces. The electrokinetic slip-velocity together with the thin electrical double layer assumption is applied on the surfaces. The position and velocity of the particles and the resulting electric and flow fields are obtained and the physical insight for the behavior of the colloidal cylinders are discussed in conjunction with the experimental observations in the literature.