Coupled Boundary Element Research Articles

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.

On the use of a tailored fluid-fluid Green's function to predict scattering from two-phase fluid interfaces

In naval applications, the precise knowledge of the acoustical behaviour of two-phase fluids is of interest. Indeed, they appear in many configurations as bubbles curtains, cavitation (along propellers and pipes) or two-phase turbulent boundary layers and wakes (gas exhaust). To model this problem, we focus on a Boundary Element/ Boundary Element coupling for low to moderate frequencies. We propose two approaches: (a) solving the Helmholtz equations in each phase by introducing the free field Green's function and (b) determining a tailored Green's function, taking into account the presence of the two-phase fluid. The determination of a tailored Green's function has two main advantages: (i) it allows a reduction of the numerical model, since it contains all the information on the fluid-fluid coupling, in particular the transmission conditions and resonances and (ii) it is source-independent and thus it gives directly the answer to any source. Numerical tests on the evaluation of radiated noise are performed to determine the efficiency of the approach based on a tailored Green's function compared to the free-field based one.

Numerical study on dynamic analysis of a nine-module floating aquaculture platform under irregular waves

A numerical study of the dynamics of a nine-module floating aquaculture platform in irregular waves is proposed, to examine the effect of the connector type between net cages and incident wave direction on the hydrodynamic response. A numerical model based on the coupled boundary element method and lumped-mass model was used to simulate the dynamic behavior of the aquaculture platform. Time-domain statistical analysis, and time-frequency analysis based on Hilbert-Huang transform on the motion responses of net cages and the mooring force responses of the platform were then carried out. The results indicate that the mooring system has a decisive influence on the frequency-domain characteristics of surge motion, while the influence of connector can be ignored. The surge motion under flexible hawser connector is less sensitive to wave incident direction, while the wave direction has a greater impact on surge motion under spherical joint connector. The amplitudes of pitch motion under two different connectors decrease significantly under oblique incident waves, compared with perpendicular incident wave condition. The wave incident direction has little effect on the overall mooring force response of the platform. The mooring force responses at different positions show completely different frequency characteristics when the cages are connected by spherical joints. The larger the pitch response of the cage, the smaller the mooring force response of the mooring chain connected to the net cage. There is a positive correlation between the gravity frequency of the mooring force response and the gravity frequency of the pitch response of the net cage.

A spectral coupled boundary element method for the simulation of nonlinear surface gravity waves

The challenge of simulating the broad open sea with limited computational resources has long been of interest in ocean engineering research. In view of this issue, this paper proposes a fully nonlinear potential flow method named the spectral coupled boundary element method (SCBEM). By leveraging the approach of domain decomposition, SCBEM achieves significantly reduced computational cost and an order of magnitude increase in computational domain compared to the conventional boundary element method (BEM). The SCBEM encompasses the marine structure with only a tiny BEM domain and employs a high-order spectral layer to simulate the broad water outside the BEM domain. The performance of the SCBEM is evaluated through comparison with the wave damping approach and literature results for regular waves, modulated wave trains, focused waves, and diffraction of a vertical cylinder. The numerical results demonstrate the effectiveness and accuracy of the SCBEM in simulating a wide range of wavelengths and nonlinear wave interactions.

Stochastic Vibrations of a System of Plates Immersed in Fluid Using a Coupled Boundary Element, Finite Element, and Finite Difference Methods Approach.

The main objective of this work is to investigate the natural vibrations of a system of two thin (Kirchhoff-Love) plates surrounded by liquid in terms of the coupled Stochastic Boundary Element Method (SBEM), Stochastic Finite Element Method (SFEM), and Stochastic Finite Difference Method (SFDM) implemented using three different probabilistic approaches. The BEM, FEM, and FDM were used equally to describe plate deformation, and the BEM was applied to describe the dynamic interaction of water on a plate surface. The plate's inertial forces were expressed using a diagonal or consistent mass matrix. The inertial forces of water were described using the mass matrix, which was fully populated and derived using the theory of double-layer potential. The main aspect of this work is the simultaneous application of the BEM, FEM, and FDM to describe and model the problem of natural vibrations in a coupled problem in solid-liquid mechanics. The second very important novelty of this work is the application of the Bhattacharyya relative entropy apparatus to test the safety of such a system in terms of potential resonance. The presented concept is part of a solution to engineering problems in the field of structure and fluid dynamics and can also be successfully applied to the analysis of the dynamics of the control surfaces of ships or aircraft.

Isogeometric FEM-BEM Coupling for Magnetostatic Problems Modeling Using Magnetic Scalar Potential

The main advantage of isogeometric analysis resides in its ability to represent exactly a wide range of geometries, and has proven great efficiency in mechanical problems compared to standard finite elements. The application of the isogeometric context to electromagnetic problems leads to the isogeometric representation of air region, a particularly ineffective process. To overcome this hindrance, in a magnetostatic context, an original magnetic scalar potential Finite element - Boundary element coupling is presented. Numerical considerations and implementation specificities are discussed, and the efficiency of the method is demonstrated.

Efficient spectral coupled boundary element method for fully nonlinear wave–structure interaction simulation

Accurately analyzing wave–structure interactions is crucial for the design and operational safety of ships and marine structures. This paper presents a fully nonlinear potential-flow approach for simulating wave–structure interactions using the newly proposed spectral coupled boundary element method (SCBEM). The SCBEM efficiently models an extensive water body that encompasses structures by establishing a boundary element method (BEM) computational domain solely around the object of interest while accurately simulating the far-field broad water by a spectral layer. To further improve efficiency, graphics processing unit acceleration is hired during iterative solving of the boundary value problem in the already small-sized interior BEM domain. Simulations are conducted to validate the accuracy of the method on cases with strong nonlinear phenomena, including wave run-up on a single cylinder, diffraction of a four-cylinder array, near-trapped modes for closely spaced columns, and gap resonance that occurred in side-by-side offloading. The wave run-up, diffraction wave pattern, near-trapped mode, and gap resonance frequency obtained by the proposed method are in good agreement with data from experiments and published literature. The quite good accuracy and the exceptional computational efficiency of the SCBEM demonstrate its promising potential for more application in practical marine problems.

Developments on the Stability of the Non-symmetric Coupling of Finite and Boundary Elements

Abstract We consider the non-symmetric coupling of finite and boundary elements to solve second-order nonlinear partial differential equations defined in unbounded domains. We present a novel condition that ensures that the associated semi-linear form induces a strongly monotone operator, keeping track of the dependence on the linear combination of the interior domain equation with the boundary integral one. We show that an optimal ellipticity condition, relating the nonlinear operator to the contraction constant of the shifted double-layer integral operator, is guaranteed by choosing a particular linear combination. These results generalize those obtained by Of and Steinbach [Is the one-equation coupling of finite and boundary element methods always stable?, ZAMM Z. Angew. Math. Mech. 93 (2013), 6–7, 476–484] and [On the ellipticity of coupled finite element and one-equation boundary element methods for boundary value problems, Numer. Math. 127 (2014), 3, 567–593], and by Steinbach [A note on the stable one-equation coupling of finite and boundary elements, SIAM J. Numer. Anal. 49 (2011), 4, 1521–1531], where the simple sum of the two coupling equations has been considered. Numerical examples confirm the theoretical results on the sharpness of the presented estimates.

Coupling of Finite and Boundary Elements for Singularly Nonlinear Transmission and Contact Problems

Abstract This article discusses the well-posedness and error analysis of the coupling of finite and boundary elements for interface problems in nonlinear elasticity. It concerns 𝑝-Laplacian-type Hencky materials with an unbounded stress-strain relation, as they arise in the modelling of ice sheets, non-Newtonian fluids or porous media. We propose a functional analytic framework for the numerical analysis and obtain a priori and a posteriori error estimates for Galerkin approximations to the resulting boundary/domain variational inequality.

Numerical modelling of the poto-poto by coupling of finite elements and boundary elements for acoustic characterization

Sustainable development encourages the construction of buildings based from local raw materials because they are ecological, renewable, available, low energy consumption, less costly, low pollution, and therefore without risk to human health. The poto-poto is a widespread material, made of soil material and bamboo, two natural materials widely used in construction throughout the world and particularly in Central African countries, due to their affordable cost or the low technical requirements for their implementation. This material (poto-poto) although widely used, remains very insufficiently explored as well from the point of view of its mechanical characteristics as from the normative point of view necessary for its modernization. This is why, given the extent and large scale of its use, we considered it important to explore it by establishing its physical characteristics. In this paper, we have chosen to continue the characterization of poto-poto already started in the previous paper by also addressing its acoustic characteristics.Indeed, the effects of climate change are manifested by abnormal variations in temperature against seasons. Hence the urgency to master the physical properties in order to make the most of it both in terms of acoustic implementation and/or thermal insulation as regards its use as a green and modern construction material. In this work, we have, on the basis of the Biot model, carried out a 2D numerical modelling of a poto-poto wall by coupling finite elements method (FEM) and boundary elements method (BEM). The MATLAB software was used to obtain the sound absorption curves of our model. We then made several specimens of our eco-material (3 cm, 4 cm and 5 cm) dosed with 2%, 4% and 6% of bamboo fibers, on which Kundt tube measurements were performed to determine the experimental sound absorption curves. The comparison of these curves with those of the model showed acceptable standard deviations and above all allowed us to conclude that the poto-poto presents an acoustic absorption in the range of frequencies 500–1250 Hz and can thus be effectively used for the acoustic implementation of the walls and certain rooms of the buildings.

A multiscale method coupling peridynamic and boundary element models for dynamic problems

A novel multiscale method coupling peridynamics (PD) and boundary element method (BEM) is proposed for dynamic problems. In the coupling model, the discontinuous behaviors of materials such as crack development and branching processes are simulated by PD while the boundary and infinite domain are modeled by time-domain BEM, which can take full advantage of their salient features. A general stiffness matrix is introduced in the coupling model to combine the PD and BEM governing equations in such a way that the continuity of displacement and the equilibrium of tractions at the PD-BEM interface are satisfied. A variable horizon method is developed to further eliminate boundary effects and ghost forces at the coupling interface between PD and BEM. Then, a concurrent coupling procedure of PD and BEM is established and its implementation is also introduced based on the dynamic explicit algorithm, i.e. the central difference method. The proposed coupling method is verified by comparing its predictions with those from analytical and experimental results. Numerical examples demonstrate the applicability of the proposed model for wave propagation, infinite domain and fracture problems under dynamic loads.

Study on Influential Mechanism of Trailing Edge Sweep Angle on Aerodynamic Noise of a Centrifugal Air Compressor

As the main noise source in the hydrogen fuel cell system, the noise level of the centrifugal air compressor greatly affects the comfort of the hydrogen fuel cell vehicle, and can be effectively reduced by optimizing the trailing edge sweep angle of the blade. In this paper, the computational fluid dynamics model was used to study the influence of the trailing edge sweep angle on the aerodynamic performance and flow characteristics of a centrifugal air compressor for vehicle fuel cells. The Ffowcs Williams–Hawkings equation and the computational fluid dynamics–boundary element coupling method were adopted to calculate the dipole source strength on the surface of the blade and the radiated aerodynamic noise, respectively, under the different trailing edge sweep angles. The results showed that the trailing edge sweep could lead to an increase in pressure ratio as well as isentropic efficiency, and a decrease in the intensity of flow separation. Meanwhile, the sound pressure level of the compressor under each working condition could be effectively reduced by the trailing edge sweep. When the rotation speed was 80,000 r·min−1 and the blade trailing edge sweep angle was 15°, the sound pressure level of the radiated aerodynamic noise was 5.8 dBA lower than that without sweep.

Numerical modeling of injection-induced earthquakes based on fully coupled thermo-poroelastic boundary element method

In recent years, there has been a substantial increase in the induced seismicity associated with geothermal systems. However, understanding and modeling of injection-induced seismicity have still remained as a challenge. This paper presents a two-dimensional fully thermo-hydro-mechanical (THM) coupled boundary element approach to characterize the fault response to forced fluid injection and assess the effect of different injection protocols on seismic risk mitigation as well as permeability enhancement. The laboratory-derived rate-and-state friction law was used to capture the frictional paradigm observed in mature faults produced in granite rocks. All phases of stick-slip cycles, including aseismic slip, propagation of dynamic rupture, and interseismic periods, were simulated. The modeling results showed that the residual values of effective normal stress and static shear stress after a particular event completely dominate the constitutive behavior of fault friction during the next seismic event. The seismic energy analyses indicated that there is a negative correlation between the seismic magnitude and the total injected volume, such that a prolonged monotonic injection eventually results in the steady slip, rather than the seismic slip. Several fluid injection protocols were designed based on a volume-controlled (VC) approach and traffic light systems (TLS) to explore their effectiveness on the seismic risk mitigation and permeability enhancement. The results showed that cyclic injection based on TLS is the most effective approach for irreversible permeability enhancement of faults through promoting slow and steady slips. Our numerical simulations also revealed that fluid extraction (backflow-fixing bottom hole pressure at atmospheric pressure), regardless of the injection style, can considerably reduce the seismicity-related risks by preventing the fast-accelerated fracture slip during the post-injection stage. This study presents novel insights into modeling the rate-and-state governed faults exposed to forced fluid injection, and provides useful approaches for shear stimulation of faults with reduced seismic risks.

A Hybridizable Discontinuous Galerkin and Boundary Element coupling method for electromagnetic simulations

In this paper, the hybridizable discontinuous Galerkin (HDG) method is combined with the electric field integral equation (EFIE) for the numerical simulation of time-harmonic Maxwell’s equation. The key feature of HDG-EFIE is that the HDG method and the boundary element method (BEM) can be coupled naturally through the extra hybrid variable on the faces of elements, the combined method produces a linear system in term of the degrees of freedom of the extra hybrid variable only. A numerical solution was compared to the analytical solution and found to be in excellent agreement.

Isogeometric FE-BE method with non-conforming coupling interface for solving elasto-thermoviscoelastic problems

The isogeometric finite element (FE) and boundary element (BE) coupling method with non-conforming interface is applied to solve 3D elasto-thermoviscoelastic problems. The thermoviscoelastic constitutive relation is expressed by Stieltjes integral, and the equivalent integral weak form of the problem is derived according to the principle of virtual work, in which the time integral is discretized by the trapezoidal difference approximation formula. The parameters of viscoelastic material are described by Prony series, and the discrete formula can be simplified by recursive relationship to avoid storing all solutions in time steps. The equilibrium and compatibility conditions of FE and BE subdomains on the coupling interface are used to couple two different system matrices, and the coupling of non-conforming non-uniform rational B-splines (NURBS) surfaces in different subdomains is achieved by using virtual knot insertion technique. In addition, the time-temperature equivalent principle is applied to observe the mechanical behavior of viscoelastic materials over a wide temperature range in a short time. Through several numerical examples, the effects of different working conditions and geometrical parameters of combustion chamber on the structure of the solid rocket motor (SRM) are studied, and the correctness and robustness of this method are verified.

Coupled stretching-bending boundary element analysis for unsymmetric magneto-electro-elastic laminates with multiple holes, cracks and inclusions

Based upon the corresponding boundary element method (BEM) for the elastic composite laminates under coupled stretching-bending (CSB) deformation, the BEM for magneto-electro-elastic (MEE) laminates is developed in this study. By using the Stroh-like complex variable formalism, recently we derived Green's functions for the MEE laminates with or without holes/cracks/inclusions. Its associated fundamental solutions employed in BEM for the MEE laminated plates under CSB deformation are then derived accordingly, with which no meshes are required for the boundaries of holes/cracks/inclusions. At each corner of the plate, three nodes are usually suggested to cover all the nodal tractions (including corner force) at, behind and ahead of the corner. If the triple nodes are placed at the same position, the auxiliary equations would be necessary to replace the trivial equations for displacement-prescribed corner and are therefore constructed in this study. Since the proposed MEE-BEM is purposely organized in the same form as its corresponding elastic case, the associated techniques in dealing with singular integrals, complete solutions for boundary and internal nodes, dual coordinates for inclined holes/cracks/inclusions, and evaluation of stress intensity factors are now successfully extended from elastic to MEE laminates. Moreover, the subregion technique is employed to cover the problems with multiple holes, cracks, and inclusions.

Numerical and experimental investigation of vibro-acoustic characteristics of a submerged stiffened cylindrical shell excited by a mechanical force

In this paper, the direct testing method is adopted to measure the underwater vibration and sound radiation of a typical stiffened cylindrical shell structure under broadband excitation. The accuracy of the test result is verified by comparing with those obtained by numerical finite element (FE)/boundary element (BE) coupling method. In addition, the free vibration, forced vibration and sound radiation characteristics of stiffened submerged cylindrical shell are presented. The testing procedures and operation steps are established to provide support for the control of the vibration and sound radiation for underwater vehicle in this study. The results show that the trend of test result curve is in good agreement with FE/BE coupling method result, however, the acceleration and sound pressure is slightly different at the resonance peak value between experimental and numerical result in 500–2500 Hz band. In addition, the high frequency vibration and sound radiation response of the structure are mainly caused by a mechanical force.

Numerical Modeling of Injection-Induced Earthquakes Based on Fully Coupled Thermo-Poroelastic Boundary Element Method

Numerical Modeling of Injection-Induced Earthquakes Based on Fully Coupled Thermo-Poroelastic Boundary Element Method

A 3D BEM-FEM approach using layered transversely isotropic half-space Green's functions in the frequency domain for SSI analyses

In this paper, a numerical approach based on a three-dimensional (3D) standard coupled boundary element method-finite element method (BEM-FEM) formulation in the frequency domain is presented. The approach allows studying the dynamics response of a structure bonded to the surface of a layered transversely isotropic half-space subjected to time-harmonic loading. The FEM is used to model the structure, and the BEM is employed for the soil, using layered half-space Green's functions, by which the radiation condition is implicitly satisfied. The BEM procedure is improved by implementing exponentially gradient (EG) elements to capture the singular behavior of tractions at the edges and corners of the soil-structure interface. The proposed approach is verified by comparison with existing semi-analytical and numerical results. Afterward, the dynamic response of a building on a layered transversely isotropic half-space subjected to lateral loading is studied, and the effect of soil layering on the response of the structure and interface tractions is demonstrated.

Experimental and mixed analytical–numerical studies for free and forced vibrations of Z-reinforced sandwich plates stiffened by steel ribs

Free and forced vibrations of Z-reinforced sandwich plates stiffened by steel ribs are researched by experimental and mixed analytical–numerical techniques. A test sample of the plate is firstly manufactured and an underwater vibration test is conducted with a random white-noise force. Meanwhile, a mixed analytical–numerical method (MA-NM) is proposed, which uses an analytical homogenization for the Z-reinforced core and coupling finite element and boundary element (FE/BE) analyses for the plate. Natural frequencies given by the MA-NM are compared with ones from the detailed FE model. Forced vibration responses from the MA-NM are validated against the experimental data. Good agreements are obtained and the high accuracy of the MA-NM is demonstrated. Parameters influences are then analyzed through the MA-NM. The core material shows a larger effect on natural frequencies than the reinforcement material. Ribs at the orthotropic midlines greatly enhance the bend stiffness of the whole structure, and ribs at the surrounding sides mainly influence boundary conditions. When in-plane coordinates of the external force near the plate center, vibration responses have fewer resonance peaks and lower amplitudes, while the influence of the thickness direction coordinate could be ignored. The fluid loading reduces both natural frequencies and vibration responses.