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Register a new userTuning and Optimizing the Finite Element Analysis with Elements of Large Nodal DOF on a Linux Cluster
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The finite element analysis of high frequency vibrations of quartz crystal plates is a necessary process required in the design of quartz crystal resonators of precision types for applications in filters and sensors. The anisotropic materials and extremely high frequency in radiofrequency range of resonators determine that vibration frequency spectra are complicated with strong couplings of large number of different vibration modes representing deformations which do not appear in usual structural problems. For instance, the higher-order thickness-shear vibrations usually representing the sharp deformation of thin plates in the thickness direction, expecting the analysis is to be done with refined meshing schemes along the relatively small thickness and consequently the large plane area. To be able to represent the precise vibration mode shapes, a very large number of elements are needed in the finite element analysis with either the three-dimensional theory or the higher-order plate theory, although considerable reduction of numbers of degree-of-freedom (DOF) are expected for the two-dimensional analysis without scarifying the accuracy. In this paper, we reviewed the software architecture for the analysis and demonstrated the evaluation and tuning of parameters for the improvement of the analysis with problems of elements with a large number of DOF in each node, or a problem with unusually large bandwidth of the banded stiffness and mass matrices in comparison with conventional finite element formulation. Such a problem can be used as an example for the optimization and tuning of problems from multi-physics analysis which are increasingly important in applications with excessive large number of DOF and bandwidth in engineering.
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An efficient method for the calculation of ferromagnetic resonant modes of magnetic structures is presented. Finite-element discretization allows flexible geometries and location dependent material parameters. The resonant modes can be used for a semi-analytical calculation of the power spectral density of the thermal white-noise, which is relevant for many sensor applications. The proposed method is validated by comparing the noise spectrum of a nano-disk with time-domain simulations.
We present a multigrid scheme for the solution of finite-element Hartree-Fock equations for diatomic molecules. It is shown to be fast and accurate, the time effort depending linearly on the number of variables. Results are given for the molecules LiH, BH, N_2 and for the Be atom in our molecular grid which agrees very well with accurate values from an atomic code. Highest accuracies were obtained by applying an extrapolation scheme; we compare with other numerical methods. For N_2 we get an accuracy below 1 nHartree.
We present an arbitrary-order spectral element method for general-purpose simulation of non-overturning water waves, described by fully nonlinear potential theory. The method can be viewed as a high-order extension of the classical finite element method proposed by Cai et al (1998) cite{CaiEtAl1998}, although the numerical implementation differs greatly. Features of the proposed spectral element method include: nodal Lagrange basis functions, a general quadrature-free approach and gradient recovery using global $L^2$ projections. The quartic nonlinear terms present in the Zakharov form of the free surface conditions can cause severe aliasing problems and consequently numerical instability for marginally resolved or very steep waves. We show how the scheme can be stabilised through a combination of over-integration of the Galerkin projections and a mild spectral filtering on a per element basis. This effectively removes any aliasing driven instabilities while retaining the high-order accuracy of the numerical scheme. The additional computational cost of the over-integration is found insignificant compared to the cost of solving the Laplace problem. The model is applied to several benchmark cases in two dimensions. The results confirm the high order accuracy of the model (exponential convergence), and demonstrate the potential for accuracy and speedup. The results of numerical experiments are in excellent agreement with both analytical and experimental results for strongly nonlinear and irregular dispersive wave propagation. The benefit of using a high-order -- possibly adapted -- spatial discretization for accurate water wave propagation over long times and distances is particularly attractive for marine hydrodynamics applications.
We introduce a textit{non-modal} analysis technique that characterizes the diffusion properties of spectral element methods for linear convection-diffusion systems. While strictly speaking only valid for linear problems, the analysis is devised so that it can give critical insights on two questions: (i) Why do spectral element methods suffer from stability issues in under-resolved computations of nonlinear problems? And, (ii) why do they successfully predict under-resolved turbulent flows even without a subgrid-scale model? The answer to these two questions can in turn provide crucial guidelines to construct more robust and accurate schemes for complex under-resolved flows, commonly found in industrial applications. For illustration purposes, this analysis technique is applied to the hybridized discontinuous Galerkin methods as representatives of spectral element methods. The effect of the polynomial order, the upwinding parameter and the Peclet number on the so-called textit{short-term diffusion} of the scheme are investigated. From a purely non-modal analysis point of view, polynomial orders between $2$ and $4$ with standard upwinding are well suited for under-resolved turbulence simulations. For lower polynomial orders, diffusion is introduced in scales that are much larger than the grid resolution. For higher polynomial orders, as well as for strong under/over-upwinding, robustness issues can be expected. The non-modal analysis results are then tested against under-resolved turbulence simulations of the Burgers, Euler and Navier-Stokes equations. While devised in the linear setting, our non-modal analysis succeeds to predict the behavior of the scheme in the nonlinear problems considered.
The enrichment formulation of double-interpolation finite element method (DFEM) is developed in this paper. DFEM is first proposed by Zheng emph{et al} (2011) and it requires two stages of interpolation to construct the trial function. The first stage of interpolation is the same as the standard finite element interpolation. Then the interpolation is reproduced by an additional procedure using the nodal values and nodal gradients which are derived from the first stage as interpolants. The re-constructed trial functions are now able to produce continuous nodal gradients, smooth nodal stress without post-processing and higher order basis without increasing the total degrees of freedom. Several benchmark numerical examples are performed to investigate accuracy and efficiency of DFEM and enriched DFEM. When compared with standard FEM, super-convergence rate and better accuracy are obtained by DFEM. For the numerical simulation of crack propagation, better accuracy is obtained in the evaluation of displacement norm, energy norm and the stress intensity factor.
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