Order Finite Element Research Articles

Numerical analysis of a nonmonotone dynamic contact problem of a non-clamped piezoelectric viscoelastic body

<p style='text-indent:20px;'>We consider a contact process between a body and a foundation. The body is assumed to be viscoelastic and piezoelectric and the contact is dynamic. Unlike many related papers, the body is assumed to be non-clamped. The contact conditions has a form of inclusions involving the Clarke subdifferential of locally Lipschitz functionals and they have nonmonotone character. The problem in its weak formulation has a form of two coupled Clarke subdifferential inclusions, from which the first one is dynamic and the second one is stationary. The main goal of the paper is numerical analysis of the studied problem. The corresponding numerical scheme is based on the spatial and temporal discretization. Furthermore, the spatial discretization is based on the first order finite element method, while the temporal discretization is based on the backward Euler scheme. We show that under suitable regularity conditions the error between the exact solution and the approximate one is estimated in an optimal way, namely it depends linearly upon the parameters of discretization.

Accurate numerical solutions of 2-D elastodynamics problems using compact high-order stencils

A new approach for the increase in the order of accuracy of high-order numerical techniques used for 2-D time-dependent elasticity (structural dynamics and wave propagation) has been suggested on uniform square and rectangular meshes. It is based on the optimization of the coefficients of the corresponding semi-discrete stencil equations with respect to the local truncation error. It is shown that the second order of accuracy of the linear finite elements with 9-point stencils is optimal and cannot be improved. However, we have calculated new 9-point stencils with the second order of accuracy that yield more accurate results than those obtained by the linear finite elements. We have also developed new 25-point stencils (similar to those for the quadratic finite and isogeometric elements) with the optimal sixth order of accuracy for the non-diagonal mass matrix and with the optimal fourth order of accuracy for the diagonal mass matrix. The numerical results are in good agreement with the theoretical findings as well as they show a big increase in accuracy of the new stencils compared with those for the high-order finite elements. At the same number of degrees of freedom, the new approach yields significantly more accurate results than those obtained by the high-order (up to the eighth order) finite elements with the non-diagonal mass matrix. The numerical experiments also show that the new approach with 25-point stencils yields very accurate results for nearly incompressible materials with Poisson’s ratio 0.4999.

Local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients

Assume that u(x) satisfies the problem Lu(x)≡−∂∂xi(aij∂u∂xj)=f(x),∀x∈Ω,u(x)=0,∀x∈∂Ω. In this article, using interpolation postprocessing technique, we will investigate the local ultraconvergence of the primal variable and the derivative of finite element approximation of u(x) using piecewise polynomials of degrees bi-k(k≥3) over a rectangular partition. Assume that k≥3 is odd and x0 is an interior vertex satisfying ρ(x0,∂Ω)≥c. Using the new interpolation postprocessing formula presented in this study, we show that the primal variable and the derivative of the post-processed finite element solution using piecewise of degrees bi-k(k≥3) at x0 converge to the primal variable and the derivative of the exact solution with order O(hk+3|lnh|) under suitable regularity and mesh conditions, respectively. Finally, we use numerical experiments to illustrate our theoretical findings.

Frequency Response Analysis of Perforated Shells with Uncertain Materials and Damage

In this paper, we give an overview of the issues one must consider when designing methods for vibration based health monitoring systems for perforated thin shells especially in relation to frequency response analysis. In particular, we allow either the material parameters or the structure or both to be random. The numerical experiments are computed using the standard high order finite element method with stochastic collocation for the cases with random material and Monte Carlo for those with damaged or random structures. The results display a wide range of responses over the experimental configurations. In perforated shell structures, the internal boundary layers can play an important role especially when damage is allowed within the penetration patterns. The computational methodology advocated here can be used to build statistical databases that are necessary for development of probabilistic damage identification methods.

Finite element simulations of second order wave resonance by motions of two bodies in a steady current

Wave resonance by forced motions of two bodies in a steady current is investigated in this paper. The work is based on second order theory of potential flow and the simulation is through a higher order finite element method with 12-node elements in time domain. Boundary conditions are imposed on mean positions of all boundaries. The i-th (i = 1,2) order potentials in the fluid domain at each time step are obtained by solving the finite element system and the spatial derivatives of the potential are calculated by differencing the shape functions directly. Numerical cases are given for two rectangular cylinders undergoing in specified oscillations in a steady current. The first and second order resonances have been captured at their resonant frequencies at different current speeds, which does not seem to have been studied yet. The numerical result shows that the current speed has important influence on the wave and force at both first and second order resonances.

Negatively curved bundles in the Igusa stable range

We use classical results in smoothing theory to extract information about the rational homotopy groups of the space of negatively curved metrics on a high dimensional manifold. It is also shown that smooth M-bundles over spheres equipped with fiberwise negatively curved metrics, represent elements of finite order in the homotopy groups of the classifying space for smooth M-bundles, provided the dimension of M is large enough.

N-torsion Groups

A group is called an n-torsion group if it has a system of defining relations of the form rn = 1 for some elements r, and for any of its finite order element a the defining relation an = 1 holds. It is assumed that the group can contain elements of infinite order. In this paper, we show that for every odd n ≥ 665 for each n-torsion group can be constructed a theory similar to that of constructed in S. I. Adian’s well-known monograph on the free Burnside groups. This allows us to explore the n-torsion groups by methods developed in that work. We prove that every n-torsion group can be specified by some independent system of defining relations; the center of any non-cyclic n-torsion group is trivial; the n-periodic product of an arbitrary family of n-torsion groups is an n-torsion group; in any recursively presented n-torsion group the word and conjugacy problems are solvable.

Comparison of different models for stress singularities in higher order finite element methods for elastic waves

AbstractIn this contribution, we present three models to capture singularities in combination with the Spectral Element Method. The first model, the continued‐fraction‐based Scaled Boundary Finite Element Method, the second model, a new approach based on enrichment with static modes, and the third model, which uses an hp‐refinement near the singularity, are compared among each other and evaluated in terms of their respective efficiency and accuracy.

Compact high-order stencils with optimal accuracy for numerical solutions of 2-D time-independent elasticity equations

A new approach for the increase in the order of accuracy of high-order numerical techniques used for time-independent elasticity is suggested on uniform square and rectangular meshes. It is based on the optimization of the coefficients of the corresponding discrete stencil equations with respect to the local truncation error. It is shown that the second order of accuracy of 2-D linear finite elements with 9-point stencils is optimal and cannot be improved. However, the order of accuracy of 25-point stencils (similar to those for quadratic finite and isogeometric elements) can be significantly improved. We have developed new 25-point stencils for 2-D elastic problems with optimal 10th order of accuracy. The numerical results are in good agreement with the theoretical findings as well as they show a big increase in accuracy of the new stencils compared with those for high-order finite elements. At the same number of degrees of freedom, the new approach yields significantly more accurate results than those obtained by high-order (up to the tenth order) finite elements. The numerical experiments also show that the new approach with 25-point stencils yields very accurate results for nearly incompressible materials with Poisson’s ratio 0.4999.

Higher order finite elements in space and time for anisotropic simulations with variational integrators. Application of an efficient GPU implementation

The efficient use of resources is one of the most important things in the current development. In the field of lightweight constructions, rubber and reinforced polymer composite materials become more and more important. This constantly increases the demands on the simulation software. The development of robust and efficient algorithms for these simulations is the aim of this paper.The application of higher order finite elements in space has a good influence on solution quality in space, but causes a higher computational cost compared to linear elements. We propose higher order finite elements in space and time in combination with a reinforced viscoelastic material formulation in a variational framework. Therefore, a higher order approximation in space and time is applied. The application of variational based time integrators guarantees the preservation of the total balance of linear momentum and the total balance of angular momentum. In order to fulfill the total balance of energy, an extension with discrete gradients is developed for the variational framework. The achieved time stepping scheme represents a very robust and consistent algorithm for the application of transient finite element simulations with reinforced viscoelastic materials and boundary conditions.In an implementation, however, the higher order approximation in space and time combined with the viscoelastic material suffers from a high computational effort. Hence, an efficient implementation is required in order to reduce the computational time to a minimum. In our approach, we face this problem by using a GPU and the programming architecture Cuda from NVIDIA, which allows a massive parallelization of time-consuming parts of the simulation. We introduce a pipeline design for the GPU implementation, which provides multiple advantages. This design allows a simple porting of an already existing implementation by means of self-managing pipeline-stages. However, a significant speedup is still achieved due to further optimizations which exploit the architecture of GPUs. In addition, when combining both hardware resources GPU and CPU the computational time can be reduced significantly once more. Therefore, our GPU implementation easily allows a distribution of computational effort between both GPU and CPU. Finally, we show in numerical examples the reached speedup of this approach, and the impact of combining the GPU and the CPU is studied in detail.

Sliding frictional contact analysis of a monoclinic coating/isotropic substrate system

Owing to innovative developments and the need to satisfy the requirements of engineering applications, new and improved materials are investigated by engineers to compensate for the drawbacks of conventional materials. There is an urgent need for the tribology community to properly investigate the behavior of these materials before utilizing them in real life. In this study, the sliding frictional contact between a monoclinic coating/isotropic substrate system and a rigid cylindrical punch is provided using an analytical formulation and finite element method (FEM). The analytical expressions for the stress and displacement fields are presented, and the resulting singular integral equations (SIEs) are solved numerically. Details of the analytical formulation and the discretization used in the solution method for the SIEs are also presented. For obtaining the numerical solution, an augmented finite element method is used with plane strain conditions and 6-node higher order finite elements. According to the results, the analytical formulation and FEM results are consistent. Using different lamina orientations, coefficient of friction, and geometric/material parameters, provides the tribology community with a highly qualified parametric study that informs surface wear and surface fatigue crack initiation in contact mechanics problems

Multi-field variational formulations and mixed finite element approximations for electrostatics and magnetostatics

In this paper we propose different multi-field variational formulations for electrostatics and magnetostatics, which can provide optimal discrete approximation of any particular vector field. The proposed formulations are constructed by appealing to mechanics point of view amenable to using general constitutive equations, which is quite different from electrostatics and magnetostatics formulations typical of physics and electrical engineering focusing on the corresponding global form suitable only for linear case. In particular, the formulations we propose can be combined with mixed discrete approximations that can ensure the continuity of tangential component of electric or magnetic field and normal component of electric displacement and magnetic flux even for low order interpolations. The choice of this kind is quite different from currently favorite choice of high order finite element interpolations used for coupling electromagnetism with mechanics. The discrete approximation is based upon Whitney’s interpolations representing the vector fields in terms of corresponding differential forms, with electric and magnetic fields as one-form and electric displacement and magnetic flux as two-form. The implementation of interpolations of this kind is made for 3D tetrahedron elements with non-standard approximation parameters defined not only at vertices (for zero-form), but at edges (for one-form) and at facets (for two-form). The results of several numerical simulations are presented to illustrate the performance of different formulations proposed herein.

Enhanced conformal perfectly matched layers for Bernstein–Bézier finite element modelling of short wave scattering

The aim of this paper is to accurately solve short wave scattering problems governed by the Helmholtz equation using the Bernstein–Bézier Finite Element method (BBFEM), combined with a conformal perfectly matched layer (PML). Enhanced PMLs, where curved geometries are represented by means of the blending map method of Gordon and Hall, are numerically investigated. In particular, the performance of radial and elliptical shaped PMLs, with a parabolic absorption function, are assessed and compared in terms of accuracy against second order Bayliss–Gunzberger–Turkel (BGT2) based local absorbing boundary conditions. Numerical results dealing with problems of Hankel source radiation and wave scattering by a rigid cylinder show that the radial shaped PML, with the h and p versions of BBFEM, enables the recovery of the predicted algebraic and exponential convergence rates of a high order finite element method (FEM). Furthermore, radial shaped BGT2 and PML have a similar performance, as long as the wave is not sufficiently well resolved. But, BGT2 performs poorly as the wave resolution increases. Additionally, the effect of harmonics of higher modes on accuracy is examined. The study reveals that the PML outperforms BGT2 for almost all propagating modes. However, a similar performance is achieved with both methods either with higher modes or a low wave resolution. Results from a multiple scattering benchmark problem provide evidence of the good performance of the proposed PMLs and the benefit of elliptical shaped PMLs in reducing significantly the size of the computational domain, without altering accuracy. The choice of the PML parameters ensuring optimal performance is also discussed.

Loss analysis of permanent magnet synchronous machines by considering circulating currents of armature windings

AbstractIn this paper, we investigate the losses caused by circulating currents generated in armature windings of permanent magnet synchronous machines by using electromagnetic field analysis. In the analysis, each parallel connected wire in the windings is modeled by finite elements in order to consider the difference in the electromotive forces and the generation of the circulating currents. The calculated losses are compared with experimental results. It is clarified that the armature copper loss in permanent magnet machines with parallel connected wires considerably increases by the circulating currents.

Computation of Composite Strengths by Limit Analysis

The paper is focused on computation of a compressive strength of composite materials by limit analysis. This method enables to determine the strength or other types of limit loads by solution of a specific optimization problem. It is also capable to predict failure zones. Abilities of the method are investigated on a particular composite -- a laboratory prepared sample consisting of a hard coal matrix and a polyurethane binder. This sample is chosen due to available CT images of the inner structure and laboratory experiments. Appropriate yield criteria are proposed for the coal and the binder in order to define the limit analysis problem. This problem is penalized and then discretized by higher order finite elements. For numerical solution, the semismooth Newton method and adaptive mesh refinements are also used. Numerical experiments in 2D for various CT scans and material parameters are performed.

A modified graded mesh and higher order finite element approximation for singular perturbation problems

A singularly perturbed convection diffusion problem is solved numerically using finite element method based on higher order polynomials. More precisely, we introduce a modified graded mesh generated using some implicitly defined functions. Higher order parameter uniform convergence is obtained in ϵ-weighted energy norm. Moreover, the error estimates obtained are optimal in the sense that they are free from logarithmic factor. A number of test examples are taken into account and a rigorous comparative analysis is presented. Moreover, we compare the proposed method with others found in the literature.

A colimit of traces of reflection groups

Li-Nadler proposed a conjecture about traces of Hecke categories, which implies the semistable part of the Betti geometric Langlands conjecture of Ben-Zvi-Nadler in genus 1. We prove a Weyl group analogue of this conjecture. Our theorem holds in the natural generality of reflection groups in Euclidean or hyperbolic space. As a corollary, we give an expression of the centralizer of a finite order element in a reflection group using homotopy theory.

Some results on FC-hypercentral units of integral group rings

We investigate the upper $$FC-$$ central series of the unit group of an integral group ring $${\mathbb Z}G$$ of a periodic group G. We prove that $${\mathcal U}={{\mathcal U}}_1({\mathbb Z}G)$$ has $$FC-$$ central height one if and only if the $$FC-$$ hypercenter of $${{\mathcal U}}_1({\mathbb Z}G)$$ is contained in the normalizer of the trivial units. Further, in these conditions, the $$FC-$$ hypercenter of the unit group is non-central if and only if G is a $$Q^{*}-$$ group. Let $$H \vartriangleleft {\mathcal U}, H$$ contained in the normalizer of the trivial units, suppose that either the elements of finite order form a subgroup or H is a polycyclic-by-finite (polycyclic) subgroup, then H is contained in the finite conjugacy center of $${{\mathcal U}}_1({\mathbb Z}G)$$ .

Groups with restrictions on proper uncountable subgroups

AbstractA groupGis called metahamiltonian if all its non-abelian subgroups are normal. The aim of this paper is to investigate the structure of uncountable groups of cardinality ℵ in which all proper subgroups of cardinality ℵ are metahamiltonian. It is proved that such a group is metahamiltonian, provided that it has no simple homomorphic images of cardinality ℵ. Furthermore, the behaviour of elements of finite order in uncountable groups is studied in the second part of the paper.

New Analysis of Galerkin FEMs for Miscible Displacement in Porous Media

The paper is concerned with optimal error estimates of classical Galerkin FEMs for the equations of incompressible miscible flows in porous media. The analysis done in the last several decades shows that classical Galerkin FEMs provide the numerical concentration of the accuracy $$O(\tau ^k+h^{r+1} + h^s)$$ in $$L^2$$ -norm. This analysis leads to the use of higher order finite element approximation to the pressure than that to the concentration in various numerical simulations to achieve the best rate of convergence. However, this error estimate is not optimal. The purpose of this paper is to establish the optimal $$L^2$$ error estimate $$O(\tau ^k + h^{r+1} + h^{s+1})$$ , from which one can see that the best convergence rate can be achieved by taking the same order $$(r=s$$ ) approximation to the concentration and pressure. Clearly Galerkin FEMs with $$r=s$$ are less expensive in computation and easier for implementation. Numerical results for both two and three-dimensional models are presented to confirm our theoretical analysis.