Fast finite element electrostatic analysis with domain decomposition method

An accurate evaluation of the dynamic responses on critical components of orthotropic bridge decks is of significance for identifying structural damage and predicting the fatigue life of long-span cable-stayed bridges. However, the traditional finite element (FE) methods are computationally cost-prohibitive for this application. In response, a new multiscale time-varying analysis method based on the dynamic balance equations and FE strategies is proposed and derived theoretically in this paper. Unlike most existing methods, the dynamic responses of this refined model is easily solved through repeated iterations, using the dynamic responses of a large-scale model as the boundary conditions. To validate the effectiveness of the method, a simply supported steel plate beam was used in a field test that demonstrated good correlation between the analytical dynamic responses and the experimental ones. To further validate this method, a case study involving a whole segment model of the Pingtan Bridge orthotropic bridge deck was established and the complex junction area between the plate and longitudinal ribs was modeled using two refinement processes. For comparison, a sufficiently refined model was also developed using ANSYS software based on the traditional FE methods. This study attempted to provide a new high-efficiency analysis framework for accurate analysis of local vibration problems. Under relatively small and easily manageable calculation conditions at each cross-scale processing level, the proposed method meets not only the requirements of global design for actual engineering applications, but also supports further in depth analysis.

A Domain Decomposition (DD) method for solving time-harmonic aeroacoustic problems is presented. The computational domain is decomposed into subdomains, and the aerodynamic-aeroacoustic boundary-value problem is solved independently for each subdomain. On the artificially introduced subdomain boundaries impedance-type transmission conditions are imposed, which assure the uniqueness of the solution. The Finite Element method is used to discretize the subdomain problems. A direct solver (LU factorization) is used to solve the subdomain systems of equations and an iterative scheme updates the transmission conditions to recover the global solution. This DD method is applied to solve the classical problem of a thin airfoil in a transverse gust. The problem is formulated in terms of the unsteady pressure, and exact non-reflecting conditions are imposed on the outer computational boundary. The accuracy of the numerical scheme is validated by comparison with existing solutions for both the near-field unsteady pressure and the farfiled radiated sound. The convergence as well as the computational time and memory requirements of the present method are investigated. It is shown that by combining the subdomain direct solvers with global iterations, this DD method significantly reduces both the computational time and the memory requirements. The present algorithm is particularly suited for parallel computing.

The non-conformal domain decomposition algorithms of the hybrid finite element-boundary integral-multilevel fast multipole algorithm are presented for computing large electromagnetic scattering and radiation problems. The numerical performance of different transmission conditions between finite element method (FEM) subdomains, and FEM and boundary integral (BI) domains is studied in the paper. It is found that the Dirichlet transmission condition employed on the interface between FEM and BI domains has faster convergence than the Robin-type transmission condition. Furthermore, the cement-element-based dual-primal finite element tearing and interconnecting (FETI-DP) domain decomposition method has faster convergence than the Lagrange-multiplier-based FETI-DP method under the FE-BI system.

A numerical implementation of the Domain Decomposition (DD) method for parallel computation of time-harmonic waves is presented. The Finite Element method on unstructured mesh is employed to discretize the boundary-value problem. A nonover-lapping mesh partitioning is used to build an overlapping domain decomposition. The algorithm also defines the new subdomain boundaries, other than the original boundaries of the global computational domain, on which transmission conditions are introduced. The independent subdomain problems can be solved concurrently, one problem per processor, on a parallel computer. The iterative scheme used to update the subdomain transmission conditions is computationally efficient and only swaps the nodal values in the overlapping region between adjacent subdomains. For exterior problems, the communication required to update the conditions on the outer boundary is substantially reduced by employing a modified Dirichlet-to-Neumann (DtN) map with truncated support. Numerical results are presented for a radiation model problem. It is shown that the DtN nonreflecting conditions eliminate spurious reflections and that the iterative scheme leads to a continuous global solution. The DD algorithm also has a very good convergence rate and significantly reduces memory requirement in comparison with a direct solver.

SummaryShear locking is a major issue emerging in the computational formulation of beam and plate finite elements of minimal number of degrees of freedom as it leads to artificial overstiffening. In this paper, discontinuous Timoshenko beam and Mindlin‐Reissner plate elements are developed by adopting the Hellinger‐Reissner functional with the displacements and through‐thickness shear strains as degrees of freedom. Heterogeneous beams and plates with weak discontinuity are considered, and the mixed formulation has been combined with the extended finite element method (FEM); thus, mixed enrichment functions are used. Both the displacement and the shear strain fields are enriched as opposed to the traditional extended FEM where only the displacement functions are enriched. The enrichment type is restricted to extrinsic mesh‐based topological local enrichment. The results from the proposed formulation correlate well with analytical solution in the case of the beam and in the case of the Mindlin‐Reissner plate with those of a finite element package (ABAQUS) and classical FEM and show higher rates of convergence. In all cases, the proposed method captures strain discontinuity accurately. Thus, the proposed method provides an accurate and a computationally more efficient way for the formulation of beam and plate finite elements of minimal number of degrees of freedom.

Coupling Mortar Finite Element and Boundary Element Methods for 2D Navier-Stokes Equations Courant Element: Before and After Iterative Methods for Solving Stiff Elliptic Problems Straight and Curved Finite Elements of Class C1 and Some Applications to Thin Shell Problems Exact Controllability to Solve the Helmholtz Equation with Absorbing Boundary Conditions Cubic Version of FEM in Elliptic Problems with Interfaces and Singularities Least Squares Mixed Finite Elements Turbulence Modelling in Finite Element Industrial Applications Necessary and Sufficient Conditions for the Numerical Approximation of a Partial Differential Equation Depending on a Small Parameter Efficient Solution Methods for Compressible Flow Computations Parallel Finite Volume Algorithms for Solving the Time-Domain Maxwell Equations on Nonstructured Meshes Coupling Between Nonlinear Maxwell and Heat Equations for an Induction Heating Problem: Modelling and Numerical Methods Solving the 3D Harmonic Maxwell Equations with Finite Elements, Lagrange Multipliers, and Iterative Methods Some Applications of the Hierarchic High Order MITC Finite Elements for Reissner-Mindlin Plates Domain Decomposition for Immiscible Displacement in Single Porosity Systems An Error Estimator for Nonconforming Approximations of a Nonlinear Problem Some Observations on Raviart-Thomas Mixed Finite Elements in p Extension for Parabolic Problems Mixed Finite Element Methods in Fluid Structure Systems A Black-Box Solver for the Solution of General Nonlinear Functional Equations by Mixed FEM A Remark on the Asymptotic Behaviour of Parabolic Variational Inequalities and Their Finite Element Approximation by the Courant Element Domain Decomposition vs. Adaptivity Material Optimization of Composites. Part Contents.

Temperature change of liquid stratification for LNG tank is more complex; therefore, it is necessary to carry out temperature field analysis of liquid stratification for LNG tank, and the orthogonal ridgelet finite element method is constructed by combining the traditional finite element method and orthogonal ridgelet transformation. Firstly, the basic characteristics of orthogonal ridgelet transform are studied. Secondly, the theory models of liquid stratification of LNG tank are studied. Thirdly, the temperature field analysis model of liquid stratification orthogonal ridgelet finite element for LNG tank is constructed. Finally, the temperature field analysis of liquid stratification of LNG tank is carried out, and the temperature changing rules of liquid stratification for LNG tank are obtained based on the traditional finite element method, B-spline wavelet finite element method, orthogonal ridgelet finite element method, and field test, and comparison of results shows that the orthogonal ridgelet finite element method has highest computing precision and efficiency. In addition, the effect of leakage heat intensity on time of rolling caused by liquid stratification of LNG tank is obtained.

A non-overlapping and non-conformal domain decomposition method (DDM) is presented for modelling large finite antenna arrays. There are two major ingredients in the proposed DDM: (a) A new second-order transmission condition is introduced, which improves convergence of the iterative process. In contrast to previous high order interface conditions, the new condition uses two second-order transverse derivatives to address the slow convergence issue of both TE and TM evanescent modes. Numerical experiments demonstrate that the convergence of the proposed algorithm is quite insensitive to the size of array. (b) The proposed non-conformal DDM not only permits the use of completely independent discretization for each of the sub-domains, but also allows adjacent sub-domains to be geometrically non-conformal. The benefits of the non-conformal nature of the proposed DDM will be fully enjoyed by a large-scale problem of practical interest, which is a 50 by 50 ultra wide band (UWB) array in the presence of a slot frequency selective surface (FSS). Numerical results verify the effectiveness of the proposed method.

We investigate a domain decomposition method (DDM) of finite element method (FEM) using Intel's many integrated core (MIC) architecture in order to determine the most effective MIC usage. For this, recently introduced high-scalable parallel method of DDM is first introduced with a detailed procedure. Then, the Intel's Xeon Phi MIC architecture is presented to understand how to apply the parallel algorithm into a multicore architecture. The parallel simulation using the Xeon Phi MIC has an advantage that traditional parallel libraries such as the message passing interface (MPI) and the open multiprocessing (OpenMP) can be used without any additional libraries. We demonstrate the DDM using popular libraries for solving linear algebra such as the linear algebra package (LAPACK) or the basic linear algebra subprograms (BLAS). Moreover, both MPI and OpenMP are used for parallel resolutions of the DDM. Finally, numerical parallel efficiencies are validated by a two-dimensional numerical example.

The method of constructing any scale wavelet finite element (WFE) based on the one-dimensional or two-dimensional Daubechies scaling functions was presented, and the corresponding WFE adaptive lifting algorithm was given. In order to obtain the nested increasing approximate subspaces of multiscale finite element, the Daubechies scaling functions with the properties of multi-resolution analysis were employed as the finite ele-ment interpolating functions. Thus, the WFE could adaptively mesh the singularity domain caused by local cracks, which resulted in better approximate solutions than the traditional finite element methods. The calculations of natural frequencies of cracked beam were used to check the accuracy of given methods. In addition, the results of cracked cantilever beam and engineering application were satisfied. So, the current methods can provide effective tools in the numerical modeling of the fault prognosis of incipient crack.

The distribution of lubricant flow field inside the two‐stage transmission reducer is very complicated during splash lubrication, which is difficult to be visually simulated and analysed using traditional finite element methods (FEM). There are many problems in model processing, algorithm selection, mesh division, and computational workload. Through sufficient investigation and experimental validation based on literature, the moving particle semi‐implicit (MPS) method is proposed to study the splash lubrication characteristics of a rail vehicle reducer. Through this method, the visual simulation calculation of multiple working conditions of reducer with complex structure is realised, and the unreasonable structure of the gearbox body is optimised. The model of oil supply demand in the Hertz contact zone of gear transmission is established, and the time‐domain variation law of oil particles number in the contact zone and oil supply state are analysed. It is found that the optimised reducer model can meet the demand of oil supply under the four typical working conditions of the rail vehicle. The higher the initial oil volume is, the more sufficient the oil supply is. Compared with the working temperature conditions of 20 and 80 °C, the lubrication effect is better at 40 and 60 °C. By analysing the power loss proportion of each gear, it is found that the sum of power loss of gear 1 and gear 2 is dominant under different working conditions, reaching 76%–99.5% of the total churning power losses.

In many experimentally realized applications, e.g. photonic crystals, solar cells and light-emitting diodes, nanophotonic systems are coupled to a thick substrate layer, which in certain cases has to be included as a part of the optical system. The finite element method (FEM) yields rigorous, high accuracy solutions of full 3D vectorial Maxwell's equations1 and allows for great flexibility and accuracy in the geometrical modelling. Time-harmonic FEM solvers have been combined with Fourier methods in domain decomposition algorithms to compute coherent solutions of these coupled system.2, 3 The basic idea of a domain decomposition approach lies in a decomposition of the domain into smaller subdomains, separate calculations of the solutions and coupling of these solutions on adjacent subdomains. In experiments light sources are often not perfectly monochromatic and hence a comparision to simulation results might only be justified if the simulation results, which include interference patterns in the substrate, are spectrally averaged. In this contribution we present a scattering matrix domain decomposition algorithm for Maxwell's equations based on FEM. We study its convergence and advantages in the context of optical simulations of silicon thin film multi-junction solar cells. This allows for substrate lighttrapping to be included in optical simulations and leads to a more realistic estimation of light path enhancement factors in thin-film devices near the band edge.

Domain decomposition for finite elements and their implementation on the IBM 3090/600 VF

A numerical method based on modified scalar finite element method (SC-FEM) is presented and programmed on MATLAB platform for optical fiber modeling purpose. We have estimated the dispersion graph, mode cut off condition, and group delay and waveguide dispersion for highly complicated chirped type refractive index profile fiber. The convergence study of our FEM formulation is carried out with respect to the number of division in core. It has been found that the numerical error becomes less than 2 % when the number of divisions in the core is more then 30. To predict the accurate waveguide dispersion characteristics, we need to compute expression $$\frac{{{\text{d}}^{2} \left( {\text{vb}} \right)}}{{{\text{dv}}^{2} }}$$ numerically by the FEM method. For that the normalized propagation constant b (in terms of $$\beta$$ ) should be an accurate enough up to around 6 decimal points. To achieve this target, we have used 1 million sampling points in our FEM simulations. Further to validate our results we have derived the higher order polynomial expression for each case. Comparison with other methods in calculation of normalized propagation constant is found to be satisfactory. In traditional FEM analysis a spurious solution is generated because the functional does not satisfy the boundary conditions in the original waveguide problem, However in our analysis a new term that compensate the missing boundary condition has been added in the functional to eliminate the spurious solutions. Our study will be useful for the analysis of optical fiber having varying refractive index profile.

In this paper, we propose an effective finite element method (FEM) on the basis of domain decomposition method. In the domain decomposition method, the domain of structure is decomposed into subdomains, which are connected each other with an independent interface using a penalty function method. Using this FEM, the analysis of the whole structure can be carried out by analyzing the individual subdomain. As a numerical example, the stress analysis of an aircraft structural panel repaired with bonded composite patches is made, and the efficiency and the accuracy of the FEM are verified. The FEM is also used to carry out several parametric studies to examine the effects of patch shape and size on the stress fields in the repaired structural panel and composite patches. Finally, we determine the optimum shape and size of a composite patch to endure the maximum tensile applied stress by a mathematical programming.