Boundary Element Method Solution Research Articles

Analytical gradients of first-order diffraction and radiation forces for design optimization of floating structures

Gradient-based design optimization of floating structures with many design variables requires efficient and accurate computation of hydrodynamic coefficients including wave excitation forces and their derivatives with respect to design variables. For large-volume structures or structures with many component members, the only practical method to determine diffraction and radiation forces is to apply a boundary element method solver. This work presents the first known boundary element method solver with implicit analytic derivatives, allowing for total derivative computation alongside force evaluation. Two case studies are presented: a single floating circular column and a structure with multiple square columns attached to a horizontal pontoon. Force results from both case studies agree well with reference data, while derivative results agree with the best comparisons available. Computational requirements limit the applicability of this method and suggest further work is required to refine this approach.

Fractional Boundary Element Solution for Nonlinear Nonlocal Thermoelastic Problems of Anisotropic Fibrous Polymer Nanomaterials

This paper provides a new fractional boundary element method (BEM) solution for nonlinear nonlocal thermoelastic problems with anisotropic fibrous polymer nanoparticles. This comprehensive BEM solution comprises two solutions: the anisotropic fibrous polymer nanoparticles problem solution and the nonlinear nonlocal thermoelasticity problem. The nonlinear nonlocal thermoelasticity problem solution separates the displacement field into complimentary and specific components. The overall displacement is obtained using the boundary element methodology, which solves a Navier-type problem, and the specific displacement is derived using the local radial point interpolation method (LRPIM). The new modified shift-splitting (NMSS) technique, which minimizes memory and processing time requirements, was utilized to solve BEM-created linear systems. The performance of NMSS was evaluated. The numerical results show how fractional and graded parameters influence the thermal stresses of nonlinear nonlocal thermoelastic issues involving anisotropic fibrous polymer nanoparticles. The numerical findings further reveal that the BEM results correlate very well with the finite element method (FEM) and analytical results, demonstrating the validity and correctness of the proposed methodology.

On the BEM solution of convection–diffusion type equations involving variable convective coefficients

In this study, numerical solutions of the unsteady convection–diffusion type equations of variable convective coefficients are investigated by two effective techniques alternative to direct boundary element method (BEM). That is, the domain boundary element and the dual reciprocity BEMs with the fundamental solution of convection–diffusion equation are used in space to transform the governing differential equations into equivalent integral equations while a backward finite difference scheme is utilized in time discretization. The fruitfulness of these combined techniques are shown by the implementation of the techniques for some widely investigated fluid dynamics problems. In fact, the lid-driven cavity flow is solved, and precise results are attained as benchmarks for assessing the accuracy of the aforementioned methods. Further, the application of the methods is extended for the natural convection flow which is governed mainly by the convection–diffusion type equations accompanied by the velocity components as variable convective coefficients. The obtained numerical results reveal that the domain BEM which uses the fundamental solution of convection–diffusion equation captures the characteristics and physical advancement of the fluid flow quite well at various combined values of the problem physical parameters.

An optimization approach for geometry design of multi-axis wave energy converter

The current global energy crisis necessitates a shift to renewable energy sources to mitigate climate change impacts. Wave energy emerges as a promising renewable resource to fulfill increasing energy demands. This energy can be extracted using wave energy converters (WECs), with multi-axis WECs (MA-WECs) being more effective than single-axis versions due to their capacity to harness energy from waves in various directions. The challenge lies in determining the ideal geometric design for MA-WECs, that can be tackled through multi-objective optimization (MOO) techniques. This research focuses on evaluating different MOO algorithms for the optimal geometric design of MA-WECs. To assess the structural response of different geometries and sizes, the study utilized the NEMOH boundary element method solver, aiming to maximize power output, lower the levelized cost of energy (LCOE), and optimize the geometry configuration. Findings indicate that the choice of optimization algorithm considerably influences the MA-WEC's optimal design, enhancing power efficiency, reducing device volume, and cutting costs more effectively than the initial design.

Computation of the effective permeability of 2D doubly porous materials with elliptical shaped pores by using boundary element method

In recent years, the prediction of the effective transport properties have received a great number of investigations. The present work is dedicated to determining the effective permeability of two-dimensional (2D) doubly porous materials made of an isotropic permeable solid matrix in which elliptical shaped pores of any size are embedded. At the interface between the fluid and the solid, the Beaver–Joseph–Saffman conditions are applied. To achieve this objective, the Boundary Element Method (BEM) is first elaborated in the simulation of velocity and pressure solution fields of two coupled Stokes and Darcy problems. Afterwards, with the help of this solution results, the effective permeablity of the doubly porous material under investigation can be determined. For the purpose of assessing the accuracy and convergence of the BEM solution, the results obtained for the velocity and pressure fields are compared with the ones provided by the finite element method (FEM). Finally, several numerical examples are carried out to analyze the fluid/solid interface influence, the effect of area fraction and geometrical properties of pores, such as the size and distribution of the pores within the matrix phase.

Towards High-Performance Linear Potential Flow BEM Solver with Low-Rank Compressions

The interaction of water waves with floating bodies can be modeled with linear potential flow theory and numerically solved with the boundary element method (BEM). This method requires the construction of dense matrices and the resolution of the corresponding linear systems. The cost of this method in terms of time and memory grows at least quadratically with the size of the mesh, and the resolution of large problems (such as large farms of wave energy converters) can, thus, be very costly. Approximating some blocks of the matrix with data-sparse matrices can limit this cost. While matrix compression with low-rank blocks has become a standard tool in the larger BEM community, the present paper provides its first application (to our knowledge) to linear potential flows. In this paper, we assess how efficiently low-rank blocks can approximate interaction matrices between distant meshes when using the Green function of linear potential flow. Due to the complexity of this Green function, a theoretical study is difficult, and numerical experiments are used to test the approximation method. Typical results on large arrays of floating bodies show that 99% of the accuracy can be reached with 10% of the coefficients of the matrix.

A Fast Multi-Frequency Iterative Acoustic Boundary Element Method Solver Based on a Preconditioned Accelerated Recycling Strategy

Common multi-frequency solution strategies for acoustic systems are either associated with high computational costs or resort to model reduction strategies, which potentially induce considerable approximation errors. In that context, this work proposes a fast multi-frequency solver for acoustic Boundary Element Method (BEM) analyses that combines conventional preconditioners with an accelerated recycling strategy, achieving in that way a fast and accurate iterative solution. Specifically, by approximating both the system and the related preconditioner by individual affine expressions, an upfront Galerkin projection on a global deflation basis is facilitated, therefore enabling the deployment of the accelerated recycling scheme. The latter acts by accelerating the construction of the related deflation projectors, therefore leading to a significantly faster recycling procedure. Finally, the efficiency of the global deflation basis is guaranteed through an Automatic Krylov subspaces Recycling for Deflation (AKR-D) [D. Panagiotopoulos, W. Desmet and E. Deckers, An accelerated subspaces recycling strategy for the deflation of parametric linear systems based on model order reduction, Comput. Methods Appl. Mech. Eng. 403 (2023) 115765] algorithm and thus, the cumulative number of iterations required by an iterative solver within a frequency sweep is drastically decreased. The proposed multi-frequency solver is tested on two industrially relevant examples of an interior and an exterior scattering problem, benchmarked against the traditional solution strategies.

Investigation of acoustic radiation from a sphere vibrating on the free surface of a finite depth water using a boundary element method

In this study, a boundary element method (BEM) is applied to investigate acoustic radiation from a sphere vibrating in pulsating mode on the free surface of finite or infinite depth water. Effect of the free surface is introduced by employing a half-space Green’s function. A modified version of the Helmholtz integral equation (HIE) is used to calculate acoustic radiation from the sphere vibrating in pulsating mode on the free surface. Free-terms of the HIE are calculated using two different forms of integrals and “dummy” boundary elements. Moreover, to simulate finite depth fluid medium, a chain image-source method is used to derive a waveguide Green’s function. To demonstrate applicability of the method presented, calculated acoustic pressures are compared with those by finite element method (FEM) and analytical calculations. Additionally, the effects of submergence depth and vibration frequency on acoustic radiation are investigated for infinitely deep water together with those of water depth and field point distance on acoustic radiation for finite water depth medium. The calculations show that there is a good agreement between BEM, FEM and analytical solutions. Also, it is observed that field point distance significantly affects the convergence behavior of waveguide Green’s function. Furthermore, it is noted that submergence depth, domain depth and vibration frequency have pronounce influence on radiated pressure amplitude and pressure field pattern.

A numerical solution for unsteady anisotropic diffusion reaction equation of time–space variable coefficients

Two‐dimensional transient problems for anisotropic functionally graded materials governed by a diffusion reaction equation of time–space dependent coefficients are considered. Using a transformation, the space–time dependent coefficients equation is reduced to a time‐dependent coefficients equation. Then taking a Laplace transform will result in a constant coefficients equation, which can be written in forms of boundary‐only integral equations and solved using a standard boundary element method for the solutions. These boundary element method solutions are then numerically inverse transformed to get the solutions in the time variable. Some problems of trigonometrically, exponentially and quadratically graded materials are solved. The numerical solutions obtained are quite accurate.

Noise assessment and low-noise flight path planning platform for urban air mobility

The rapidly emerging market of urban air mobility has raised considerable public concern about noise exposure and annoyance, which is likely to limit the development of future aerial transportation. This study presents a comprehensive approach to evaluate the flight noise of an urban air mobility vehicle under various operating conditions and to provide practical flight strategies for low-noise path planning. Two methods with different fidelity levels are considered and compared in evaluating sound sources. The first approach is based on an efficient semi-empirical model to predict the rotor noise and a boundary element method solver to account for the noise scattering from the vehicle fuselage. The second approach is based on high-fidelity computational aeroacoustic simulations considering a practical full-scale urban air mobility vehicle. Then, a Gaussian beam method is used to calculate the far-field sound propagation considering practical physical phenomena in complex environments, including sound refraction attenuation in the atmosphere and reflection from obstacles. A low-noise flight planning strategy is proposed using a heuristic method to optimize the vehicle flight path in realistic urban scenarios. The proposed methods should contribute to future green aerial transportation in low-altitude operations.

Roundness and slenderness effects on the dynamic characteristics of spar-type floating offshore wind turbine

Abstract Spar-type floating offshore wind turbine has been massively developed considering its design simplicity and stability to withstand the wave-induced motion. However, the variation of the local sea level and the readiness of supporting production facilities demand the spar design to adapt in a viable way. Considering this, the present article investigated how the slenderness (length over diameter ratio) and the roundness of cross section influence the hydrodynamic characteristics, which are the crucial parameters of floater performances. The OC3-Hywind spar-type floating platform was adapted as the reference model. The length of the reference floater was then varied with a ratio of 1.5, 2, 2.5, and 3 and the diameter was proportionally scaled to obtain constant buoyancy. The number of the sides which indicated the roundness of the cross section was varied to be 4, 6, 8, 10, 12, 14, and infinity (cylindrical shape). The analysis was conducted using potential flow theory in a boundary element method solver through an open-source code NEMOH. Initially, panel convergence was conducted and compared with the experimental results of the reference model to obtain the appropriate simulation settings before being used for the case configuration analysis. Results stated that the roundness effect with sides greater than 16 had little effect on dynamic characteristics. Meanwhile, the spar with the largest diameter was more stable against the translational motion.

Numerical modelling of a box-type and bottom-detached oscillating water column wave energy conversion device: a comparison with experimental data and between BEM and CFD numerical modelling

Utilization of Boundary Element Method (BEM) based on linear potential flow for modelling Oscillating Water Column (OWC) devices has gained popularity in the last two decades. The commercial BEM solver WAMIT has been used widely for modelling OWCs and validated using experimental modelling (Delauré et. al. 2003, Bingham et. al. 2015, Faÿ 2020). The open-source BEM solver NEMOH has however been mostly ineffective in modelling OWCs since the main approach adopted previously modelled the imaginary piston as a thin disk. In this research, the multi-body interaction problem has been adopted in modelling a box-type and bottom-detached OWC device in NEMOH, where the imaginary piston has been modelled to the length of the internal water column (Penalba et.al. 2017) and compared with experimental data. A further comparison is drawn with the numerical method of Computational Fluid Dynamics (CFD) , which has shown to be accurate for modelling OWC devices (Simonetti et. al. 2015), yet requires significantly higher computational resources than BEM. A two-dimensional CFD numerical wave tank, developed generating and absorbing waves with the waves2Foam toolbox (Jacobsen et al., 2012) of the open-source package OpenFOAM, is used for comparative purposes.
 REFERENCES:
 Delauré, Y. & Lewis, A. (2003). “3D hydrodynamic modelling of fixed oscillating water column wave power plant by a boundary element methods.” Ocean Engineering. 30. 309-330. 10.1016/S0029-8018(02)00032-X.
 Bingham, H.B.; Ducasse, D.; Nielsen, K. et al (2015). “Hydrodynamic analysis of oscillating water column wave energy devices.” J. Ocean Eng. Mar. Energy 1, 405–419 (2015). https://doi.org/10.1007/s40722-015-0032-4
 Faÿ F.-X. (2020). “Modelling and Control for the Oscillating Water Column in Wave Energy Conversion.” PhD Thesis, Universidad del País Vasco.
 Jacobsen, N.G., Fuhrman, D.R. & Fredsøe, J., (2012). “A wave generation toolbox for the open-source CFD library : OpenFoam”. International Journal for numerical methods in fluids, 70, pp.1073–1088.
 Simonetti, Irene & Cappietti, Lorenzo & Elsafti, Hisham & Oumeraci, Hocine. (2015). “Numerical Modelling of Fixed Oscillating Water Column Wave Energy Conversion Devices: Toward Geometry Hydraulic Optimization.” 10.1115/OMAE2015-42056.
 Penalba, M., Kelly, T., Ringwood, J. 2017, NEMOH for Modelling Wave Energy Converters: A Comparative Study with WAMIT. 12th European Wave and Tidal Energy Conference , Cork, Ireland. https://www.researchgate.net/publication/319160625.

Some Green’s functions for steady-state heat conduction in anisotropic plane media and their application to thermoelastic boundary element analysis

In this paper, we present several Green’s functions of steady-state heat conduction in anisotropic plane media, including (1) an infinite plane, (2) a half-plane, (3) a bi-material plane, (4) an infinite plane with an elliptical hole or a straight crack, and (5) an infinite plane with an elliptical elastic inclusion. These solutions are obtained by using the link between anisotropic elasticity and heat conduction. We start with reducing the Stroh formalism for two-dimensional anisotropic elasticity to anti-plane deformation and then use the analogy between anti-plane deformation and heat conduction. These Green’s functions serve as fundamental solutions of boundary element method, and the derived temperature field and gradients on the boundary are used as input for thermoelastic analysis. The results of heat conduction and thermoelasticity are verified with analytical solutions or finite element solutions.

H-Matrix BEM application and validation for space habitat

Acoustic requirements for space habitats are stringent and involve a combination of noise sources minimization and public address and general alarm (PAGA) system validation. Simulation solutions are essential for demonstrating compliance requirement as early as possible and continuously throughout the development cycle of the spacecraft. Space habitats' sound field can be computationally challenging to predict in a deterministic manner for most of the audible spectrum as the volume can be important. For this application, an initial PAGA system design assessment was performed using Ray-Tracing, a geometric statistical method, which must be validated using a deterministic method. For validation purpose, the Boundary Element Method (BEM) was chosen for this application. Given the size of the structure and the desired frequency range, the BEM model was very large, containing over 250,000 wetted nodes, and a standard BEM solver required approximately 200GB of RAM. To limit computing hardware constraints, a newly implemented commercially available H-Matrix BEM solver was used. Pressure maps of a PAGA system simulation on a generic space habitat resulting from the Ray-Tracing and H-Matrix BEM solvers are compared to a standard BEM simulation. Results are compared both in terms of accuracy and computational performance, showing the advantages of using multiple simulation methods at various stages of the design.

Development of numerical scheme using boundary element method for heat conduction problems

This paper deals with the boundary element method solution of set of parabolic differential equations of heat conducting problems with continuous and discontinuous boundary conditions. Time-dependent terms have been used as a non-homogeneous forcing function by changing a parabolic differential equation into a Poisson's equation. The suggested approach for solving the Poisson equation has been modified which decreases the number of algebraic equations and reduces the computational time. The constant coefficient matrix and LU decomposition provide an economical means for the solution of the concern problem. The discretization of the boundary of parabolic differential equations with respect to (w.r.t.) time is one of the main features of this article. Direct determination of unknown at interior points is also trademark of the proposed method. The method has been supported by examples, which include discontinuous boundary conditions and non-homogeneous forcing terms to highlight its effectiveness and accuracy.

A Potential‐Based Boundary Element Implementation for Modeling Multiple Scattering from Local and Nonlocal Plasmonic Nanowires

AbstractA novel boundary element implementation that models multiple scattering of plasmonic nanowires is presented. The modeling is based on potentials and the materials constituting the wires can be local (described by the local response model) or nonlocal (described by the nonlocal hydrodynamic model). The nonlocal hydrodynamic model (HDM) provides an important approximation describing nonclassical effects associated with the collective motion of free electrons in metals. The modeling is challenging as different interface conditions are needed at a boundary which separates 1) a local medium from a local medium; 2) a local (nonlocal) medium from a nonlocal (local) medium; and 3) a nonlocal medium from a nonlocal medium. The algorithm can address constructs of arbitrary geometry and material composition within the HDM; thus, it becomes a complete numerical tool for exploration of nonlocal nanowires. Fictitious sources are imposed at the boundaries, linking the scattered fields to the imposed sources, and matching the interface conditions at the boundaries. This procedure yields a set of Boundary Integral Equations (BIEs). Then, the BIEs are numerically solved utilizing the Boundary Element Method (BEM). The results from the BEM solver are verified quantitatively and qualitatively showing good agreement in both the near‐ and the far‐field regimes.

Isogeometric indirect BEM solution based on virtual continuous sources placed directly on the boundary of 2D Helmholtz acoustic problems

An indirect boundary element method (BEM) based on isogeometric analysis (IGA) is proposed for 2D Helmholtz acoustic problems using virtual continuous sources placed directly on the problem boundary. The virtual sources form a virtual boundary identical to the main problem boundary. The proposed solution couples the CAD model with the analysis model by approximating both the CAD model and the virtual continuous sources with the same non-uniform rational B-spline functions (NURBS). Moreover, neither domain discretization nor truncation boundaries at the far-field are required. This solution creates only one coefficient matrix by directly arranging the linear system of equations.The solution follows a collocation scheme on the boundaries based on Greville abscissae with offsets wherever C0 continuity is encountered to permit an easy prediction for the normal directions and the free-terms at the collocation points required in the cases of Neumann and Robin boundary conditions. It allows us also to treat all integrals with standard Gauss quadrature points.Several numerical examples for exterior and interior acoustic problems are discussed to verify the proposed solution with comparisons to analytical solutions and previously published numerical results. These examples prove the robustness of the proposed solution even for high wavenumbers, in contrast to the previous attempts which implemented extensive investigations to find out the optimum place of the sources outside the domain producing minimum errors. Furthermore, no fictitious eigenfrequency problem is observed for exterior acoustic problems.

A nonlinear model of multifractured horizontal wells in heterogeneous gas reservoirs considering the effect of stress sensitivity

Tight gas reservoirs are usually heterogeneous and associated with notable stress sensitivity during the production, directly influencing long-term development. This paper proposes a nonlinear model of multifractured horizontal wells in heterogeneous gas reservoirs considering stress-sensitive effects. Based on the linearization by stress-dependent pseudopressure and pseudotime functions, the boundary element method and Green's solution are applied to solve the model semianalytically. The effects of the stress sensitivity, production settings and subblock number are analyzed. The stress sensitivity is mainly reflected at the middle and later production stages, imposing a notable negative effect on bottomhole pressure, reducing the bottomhole pressure in a two-block reservoir by approximately 1%–28.2%. The recoveries in a two-block reservoir with different production settings under a terminal pressure vary from 16.3% to 42.49%, and thus a reasonable rate should be set to weaken the stress-sensitive effects on pressure loss to obtain a higher cumulative production within a given time. For a three-block reservoir, the proper increase of the fracture parameters in low permeability blocks can weaken the influences of heterogeneity and make the development more uniformly. This paper provides a theoretical basis for improving gas production performance forecasting for efficient development of natural gas.

A Parallel Boundary Element Method for the Electromagnetic Analysis of Large Structures With Lossy Conductors

In this paper, we propose an efficient parallelization strategy for boundary element method (BEM) solvers that perform the electromagnetic analysis of structures with lossy conductors. The proposed solver is accelerated with the adaptive integral method, can model both homogeneous and multilayered background media, and supports excitation via lumped ports or an incident field. Unlike existing parallel BEM solvers, we use a formulation that rigorously models the skin effect, which results in two coupled computational workloads. The external-problem workload models electromagnetic coupling between conductive objects, while the internal-problem workload describes field distributions within them. We propose a parallelization strategy that distributes these two workloads evenly over thousands of processing cores. The external-problem workload is balanced in the same manner as existing parallel solvers that employ approximate models for conductive objects. However, we assert that the internal-problem workload should be balanced by algorithms from scheduling theory. The parallel scalability of the proposed solver is tested on three different structures found in both integrated circuits and metasurfaces. The proposed parallelization strategy runs efficiently on distributed-memory computers with thousands of CPU cores and outperforms competing strategies derived from existing methods.

Joint inversion of tectonic stress and magma pressures using dyke trajectories

AbstractIn volcano-tectonic regions, dyke propagation from shallow magmatic chambers is often controlled by the interaction of the local and regional stress fields. The variations of the stress fields result from a combination of factors including the regional tectonic stress, the geometry of pressurized magma chambers, the layering and the pre-existing discontinuities (e.g. fractures). In this contribution, we describe and apply a new multiparametric inversion technique based on geomechanics that can invert for both the far field stress attributes and the internal pressure of magma chambers or stocks, constrained by observed dyke or eruptive fissure orientations. This technique is based on the superposition principle and uses linear elastic models that can be solved using many types of numerical methods. For practical reasons, we chose a 3D boundary element method (BEM) for a heterogeneous elastic half-space, where magma chambers are modelled as pressurized cavities. To verify this approach, the BEM solution has been validated against the known 3D analytical solution of a pressurized cylindrical cavity. Then the effectiveness of this technique and its practical use is demonstrated through application to natural examples of dyke network development around two different volcanic systems, the Spanish Peaks (USA) and the Galapagos Islands (Ecuador). Results demonstrate that regional stress characteristics as well as the internal pressure of magma chambers can be estimated from observed radial and circumferential dyke patterns and some knowledge of magma chamber geometry.