Coupling Of Subdomains Research Articles

Three-dimensional IBEM solution to seismic wave scattering by a near-fault sedimentary basin

In this paper, the indirect boundary element method (IBEM) is developed to simulate the three-dimensional (3-D) whole process ground motion in a near-fault complex site. The near-fault large-scale sedimentary basin is investigated as an example by establishing a 3-D kinematic finite fault model and subdomain coupling IBEM. The kinematic boundary conditions of the fault and the continuity of displacement and stress at the interface of the basin are precisely satisfied. The boundary integral equations are solved to obtain the wave fields of each subdomain. The boundary discretization is differentiated by profiling the subdomain wave field to minimize computational cost. The calculation frequency up to 5Hz is achieved on an ordinary workstation. Further, the coupling mechanism of the near-fault and basin amplification effects is discussed in-depth. The influence of the source frequency, fault dip, and fault distance on the basin amplification effect is investigated systematically. The results show that the presence of the sedimentary basin significantly amplifies the near-fault seismic response, especially in the 0 Hz-1 Hz frequency band. The displacement amplitude at the peak frequency (0.22 Hz) inside the basin is amplified 13.11 times compared to that without the basin, while the peak ground displacement (PGD), peak ground velocity (PGV) and peak ground acceleration (PGA) amplify by 1.98 times, 3.42 times and 4.98 times, respectively. The study provides an effective method for simulating 3-D ground motion fields in near-fault complex sites. The conclusions provide an essential reference for seismic risk assessment and seismic zoning in near-fault sedimentary basins.

Parametric stitching for smooth coupling of subdomains with non-matching discretizations

Incompatible parametric subdomains are common in computer-aided design (CAD) and in computer-aided engineering (CAE). In this paper, we present a unified formulation to smoothly couple non-matching parametric domains for both geometric modeling and analysis of behavior. The key concept used to accomplish the coupling is a “parametric stitching” or p-stitching interface between the incompatible patches. Specifically, p-stitching permits independently varying fields with assured, arbitrary smoothness at the interface between the coupled subdomains. The developed procedure enables modular construction of coupling problems with compatible interfaces as well as the ability to characterize sharp changes in gradient, as at dissimilar material interfaces. Fundamental to the developed methodology is enriched field approximations. The base approximations in the subdomains are enriched by the interfacial fields constructed as a function of distance from the coupling interfaces. The proposed method is argued to be the smooth extension of the dual–primal method such as the localized version of the Lagrange multiplier method. In the present paper, non-uniform rational B-splines (NURBS) are chosen for discretizing the parametric subdomains. The developed procedure though is valid for other representations of subdomains whose basis functions obey partition of unity. We validate the proposed method through patch tests and demonstrate the approach on several two- and three-dimensional elastostatic as well as heat conduction numerical examples.

Multiscale mixed methods for two-phase flows in high-contrast porous media

The Multiscale Robin Coupled Method (MRCM) is a domain decomposition procedure that has been developed to efficiently approximate velocity and pressure fields for single-phase flows in highly heterogeneous porous media. It generalizes other well-established multiscale domain decomposition mixed methods and it adds great flexibility to the choice of interface spaces as well as in the boundary conditions for subdomain coupling. We investigate the approximation of two phase flows in porous media using the MRCM to compute velocity fields. We consider an operator splitting strategy, where a scalar conservation law for the saturation of one of the phases and the velocity field are updated sequentially. We find that the coupling of nearest neighbor subdomains through the imposition of a continuous pressure (respectively, normal fluxes) is the best strategy to approximate two-phase flows in the presence of high (resp., low) permeability channels (resp., regions). A new adaptivity strategy for setting an algorithmic parameter of the MRCM, that controls the relative importance of Dirichlet and Neumann boundary conditions in the coupling of subdomains, is proposed and tested in challenging, high-contrast permeability fields. Our numerical simulations of two-phase flows show that by switching between existing multiscale procedures we can observe unprecedented accuracy, in that we produce better solutions for problems with high-contrast permeability coefficients when compared to solutions obtained with some standard multiscale mixed methods.

Moving overlapping grid methodology of spectral accuracy for incompressible flow solutions around rigid bodies in motion

The simulation of fluid flow around moving rigid bodies has proven to be a difficult task for traditional computational fluid dynamics solvers. Decomposing the global domain into overlapping subdomains and allowing each subdomain to move independently, permits solutions to many flow problems with complex moving geometries to be determined in a straightforward manner. The present development of the moving overlapping grid method is built within a spectral element method incompressible flow solver, and uses an Arbitrary Lagrangian-Eulerian formulation of the governing equations to prescribe subdomain motions. The method maintains global spectral spatial accuracy across the subdomains with the polynomial refinement. The global high-order temporal accuracy of the method is also maintained through subdomain coupling enforced by an explicit interface temporal extrapolation scheme. The method produces aerodynamic forces and vortex shedding around two- and three-dimensional moving rigid bodies that is in line with published experimental and computational data. Additionally, the method achieves near linear computational scaling to thousands of cores.

Concurrent Multi-Domain Simulations in Structural Dynamics Using Multiple Grid and Multiple Time-Scale (MGMT) Method

This paper describes a generic algorithm for concurrently solving multiple sub-domains that are selectively discretized in space and time. The mathematical background for this approach is largely based upon the fundamental principles of domain decomposition methods (DDM) and Lagrange multipliers. A proof of stability is provided using energy method and overall efficiency, accuracy and stability of multiple sub-domain coupling is evaluated using a series of numerical examples. Numerical stability is verified by ensuring energy balance at global as well as component sub-domain level. Discussed examples highlight the greatest advantage of MGMT method; which is high simulation speedups (at the cost of reasonably small errors).

Substructure/fluid subdomain coupling method for large vibroacoustic problems

Dynamic analysis of complex and large structures may be costly from a numerical point of view. For coupled vibroacoustic finite element models, the importance of reducing the size becomes obvious because the fluid degrees of freedom must be added to the structural ones. In this paper, a component mode synthesis method is proposed for large vibroacoustic interaction problems. This method couples fluid subdomains and dynamical substructuring of Craig and Bampton type. The acoustic formulation is written in terms of the velocity potential, which implies several advantages: coupled algebraic systems remain symmetric, and a potential formulation allows a direct extension of Craig and Bampton\'s method to acoustics. Those properties make the proposed method easy to implement in an existing finite element code because the local numerical treatment of substructures and fluid subdomains is undifferentiated. Test cases are then presented for axisymmetric geometries. Numerical results tend to prove the validity and the efficiency of the proposed method.

Coupling of BEM subdomains – BETI applied to collocation BEM with mixed basis functions

This work is about the coupling of subdomains for elasticity problems using the boundary element tearing and interconnecting method (BETI). BETI is a substructuring technique which is able to solve large problems involving subdomains efficiently, because its formulation is ideally suited for parallel computation. In contrast to the original formulation of BETI, where the symmetric Galerkin boundary element method (SGBEM) is applied, here the traditional collocation BEM is used. As will be shown the solution algorithm is applied in a very similar way as in the original version. Special attention will be paid to the distribution of tractions along interfaces of BEM subdomains. Especially if the coupling interface is not smooth but has corners and edges erroneous traction results can be obtained by using the non-symmetric approximation of the Dirichlet-to-Neumann map. For a sequential calculation involving several load steps this leads to unacceptable results if a classical continuous approximation of the physical fields is used. Therefore discontinuous basis functions for the displacement and traction field are introduced. As will be shown in this work the quality of results can be improved drastically by this technique.

Sub-Domain Decomposition Iterative Method Combined With ACA: An Efficient Technique for the Scattering From a Large Highly Conducting Rough Sea Surface

The sub-domain decomposition iterative method (SDIM) is presented to solve efficiently a large linear system obtained by sampling the boundary integral equations from the method of moments. Then, this new technique is tested on the electromagnetic scattering problem from a large highly conducting one-dimensional rough sea surface. The diagonal blocks are the local impedance matrices corresponding to the geometry sub-domains while the off-diagonal blocks are the coupling matrices describing the interaction between two different sub-domains. The principle of SDIM is to invert the impedance matrix by blocks reducing significantly the complexity in comparison to a direct LU inversion of the whole impedance matrix. In addition, to accelerate the matrix-vector products and to reduce the memory requirement, the adaptive cross approximation (ACA) is applied to compress the sub-domain coupling matrices. For microwave frequencies, the results show that SDIM converges rapidly and faster for the TM polarization. Moreover, the mean compression ratio of ACA is of the order of 98%, which makes this method very efficient.

Solving multizone and multicrack elastostatic problems: A fast multipole symmetric Galerkin boundary element method approach

Symmetric Galerkin boundary element methods (SGBEMs) for three-dimensional elastostatic problems give rise to fully populated (albeit symmetric) matrix equations, entailing high solution times for large models. This paper is concerned with the formulation and implementation of a multi-level fast multipole SGBEM (FM-SGBEM) for multi-zone elasticity problems with cracks. The subdomain coupling approach is based on a minimal set of interfacial unknowns (i.e. one displacement and one traction vector at any interfacial point) that are defined globally for the complete multizone configuration. Then, unknowns for each subdomain are defined in terms of the global unknowns, with appropriate sign conventions for tractions induced by subdomain numbering. This formulation (i) automatically enforces the perfect-bonding transmission conditions between subdomains, and (ii) is globally symmetric. The subsequent FM-SGBEM basically proceeds by assembling contributions from each subregion, which can be computed by means of an existing single-domain FM-SGBEM implementation such as that previously presented by the authors (Pham et al., Eng Anal Bound Elem 2012;36:1838–47 [36]). Along the way, the computational performance of the FM-SGBEM is enhanced through (a) suitable storage of the near-field contribution to the SGBEM matrix equation and (b) preconditioning by means of nested GMRES. The formulation is validated on numerical experiments for 3D configurations involving many cracks and inclusions, and of sizes up to N≈106.

Multi-directional and multi-time step absorbing layer for unbounded domain

Variant techniques are proposed for reproducing the elastic wave propagation in an unbounded medium such as the infinite elements, the absorbing boundary conditions or the perfect matched layers. Here, a simplified approach is adopted by considering absorbing layers characterized by the viscous Rayleigh matrix as studied by Semblat et al. [16] and Rajagopal et al. [14]. Here, further improvements to this procedure are provided. First, we start by establishing the strong form for the elastic wave propagation in a medium characterized by the Rayleigh matrix. This strong form will be used for deriving optimal conditions for damping out in the most efficient way the incident waves while minimizing the spurious reflected waves at the interface between the domain of interest and the Rayleigh damping layer. A procedure for designing the absorbing layer is proposed by targeting a performance criterion expressed in terms of logarithmic decrement of the wave amplitude in the layer thickness. Second, the GC subdomain coupling method, proposed by Combescure and Gravouil [9], is introduced for enabling the choice of any Newmark time integration schemes associated with different time steps depending on subdomains. When wave propagation is predicted by an explicit time integrator, the subdomain strategy is of great interest because it enables a different time integrator for the absorbing layer to be adopted. An external coupling software, based on the GC method, is used to carry out multi=time step explicit/implicit co-computations, making interact in time an explicit FE code (Europlexus) for the domain of interest, with an implicit FE code (Cast3m) handling the absorbing boundary layers. The efficiency of the approach is shown in 1D and 2D elastic wave propagation problems.

Structural‐acoustic coupling on non‐conforming meshes with quadratic shape functions

SUMMARYFully coupled finite element/boundary element models are a popular choice when modelling structures that are submerged in heavy fluids. To achieve coupling of subdomains with non‐conforming discretizations at their common interface, the coupling conditions are usually formulated in a weak sense. The coupling matrices are evaluated by integrating products of piecewise polynomials on independent meshes. The case of interfacing elements with linear shape functions on unrelated meshes has been well covered in the literature. This paper presents a solution to the problem of evaluating the coupling matrix for interfacing elements with quadratic shape functions on unrelated meshes. The isoparametric finite elements have eight nodes (Serendipity) and the discontinuous boundary elements have nine nodes (Lagrange). Results using linear and quadratic shape functions on conforming and non‐conforming meshes are compared for an example of a fluid‐loaded point‐excited sphere. It is shown that the coupling error decreases when quadratic shape functions are used. Copyright © 2012 John Wiley & Sons, Ltd.

Explicit/implicit multi-time step co-computations for blast analyses on a reinforced concrete frame structure

The paper explores the coupling between finite element codes based on implicit and explicit time integration schemes. An external coupling software has been developed based on the multi-time subdomain coupling algorithm, labelled as the GC method in the literature, allowing coupling of any time integration schemes from the Newmark family with the appropriate time step dependent on the features of the subdomain. Explicit/implicit multi-time step co-computations have been carried out for predicting the response of a reinforced concrete frame structure under a blast loading striking its front face. The well known SPEAR structure, pseudo-dynamically tested at the ELSA laboratory, in Ispra, Italy, has been adopted as a testing structure for blast analyses. The structure has been modelled with multi-fibre elements, whose steel and concrete fibres composing the beam and column section, have cyclic and non-linear behaviours. Anticipating that non-linear phenomena will essentially occur in the slender columns of the SPEAR frame structure, a mesh partitioning has been considered, composed of two subdomains: a first subdomain, with linear material behaviour, gathering beam elements handled by an implicit time integration scheme with a large time step and a second subdomain, with non-linear material behaviour, handled by an explicit time integration scheme with a fine time step. Results from co-computations with time step ratio up to 1000 have been checked with respect to full explicit computations. Despite the number of interface nodes involved into the subdomain partitioning, the explicit/implicit multi-time step co-computations provide very accurate global and local response parameters in terms of maximum inter-storey drifts, maximum mid-height column drifts or extreme strains in concrete and reinforcement, while reducing by 60% the computation time.

Coupling subdomains with heterogeneous time integrators and incompatible time steps

The work presented in this publication can be categorized among domain decomposition methods of the dual Schur type applied to structural dynamics. This approach leads to lower CPU times and better control of the accuracy of the time discretization and allows to take into account multi-time-scale effects which arise in transient structural dynamics. In order to consider incompatible time scales, one has to enforce continuity at the interfaces between the subdomains. Here, we propose a general formalism which enables the coupling of subdomains with their own numerical time integration scheme. The proposed method enables one to take into account possible nonlinearities which may present different time scale between the subdomains in a general manner for a wide range of time numerical scheme. This method also offers an important improvement for industrial software with easy implementation. Linear and nonlinear numerical examples are proposed in order to show the efficiency and the robustness of the method.

Allosteric Communication in Myosin V: From Small Conformational Changes to Large Directed Movements

The rigor to post-rigor transition in myosin, a consequence of ATP binding, plays an essential role in the Lymn–Taylor functional cycle because it results in the dissociation of the actomyosin complex after the powerstroke. On the basis of the X-ray structures of myosin V, we have developed a new normal mode superposition model for the transition path between the two states. Rigid-body motions of the various subdomains and specific residues at the subdomain interfaces are key elements in the transition. The allosteric communication between the nucleotide binding site and the U50/L50 cleft is shown to result from local changes due to ATP binding, which induce large amplitude motions that are encoded in the structure of the protein. The triggering event is the change in the interaction of switch I and the P-loop, which is stabilized by ATP binding. The motion of switch I, which is a relatively rigid element of the U50 subdomain, leads directly to a partial opening of the U50/L50 cleft; the latter is expected to weaken the binding of myosin to actin. The calculated transition path demonstrates the nature of the subdomain coupling and offers an explanation for the mutual exclusion of ATP and actin binding. The mechanism of the uncoupling of the converter from the motor head, an essential part of the transition, is elucidated. The origin of the partial untwisting of the central β-sheet in the rigor to post-rigor transition is described.

A Locking Free Mortar Method for Coupling of Fourier Models and 3D Non-Axisymetric Shell Structures

In this paper a new method is introduced for the coupling of sub-domains some of them being axisymetric and others being fully 3D. A dedicated “mortar” method is presented and a number of examples are presented for linear, buckling, and non linear geometrical and material cases. The method presented uses the COMU shell element which permits non linear computation of axisymetric structures either with non axisymetric initial imperfections or non axisymetrically loaded. This element is combined with usual 3D DKT shell element via a special mortar method.

A defect equation approach for the coupling of subdomains in domain decomposition methods

A defect equation for the coupling of nonlinear subproblems defined in nonoverlapped subdomains arise in domain decomposition methods is presented. Numerical solutions of defect equations by means of quasi-Newton methods are considered.