Fraeijs de Veubeke variational principle-based cylindrical shell finite element model

In this paper a dimensionally reduced cylindrical shell model based on the dual-mixed variational principle of Fraeijs de Veubeke will be presented. The fundamental variables of this variational principle are the not a priori symmetric stress tensor and the skew-symmetric rotation tensor. The tensor of first-order stress functions is applied to satisfy translational equilibrium. A shell model derived in this way makes the application of the classical kinematical hypotheses unnecessary, and enables us to use unmodified three-dimensional constitutive equations. On the basis of this shell model, a new dual-mixed cylindrical shell finite elements capable of both h- and p-approximation can be derived. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Kinematics of finite elastoplastic deformations

In this paper we review various approaches to the decomposition of total strains into elastic and nonelastic (plastic) components in the multiplicative representation of the deformation gradient tensor. We briefly describe the kinematics of finite deformations and arbitrary plastic flows. We show that products of principal values of distortion tensors for elastic and plastic deformations define principal values of the distortion tensor for total deformations. We describe two groups of methods for decomposing deformations and their rates into elastic and nonelastic components. The methods of the first group additively decompose specially built tensors defined in a common basis (initial, current, or “intermediate”). The second group implies a certain relation connecting tensors that describe elastic and plastic deformations. We adduce an example of constructing constitutive relations for elastoplastic continuums at large deformations from thermodynamic equations.     

Efficient algorithms for solving tensor product finite element equations

Summary There are currently several highly efficient methods for solving linear systems associated with finite difference approximations of Poisson's equation in rectangular regions. These techniques are employed to develop both direct and iterative methods for solving the linear systems arising from the use ofC ⁰ quadratic orC ¹ cubic tensor product finite elements.     

A mixed nonconforming finite element for linear elasticity

This article considers a mixed finite element method for linear elasticity. It is based on a modified mixed formulation that enforces the continuity of the stress weakly by adding a jump term of the approximated stress on interior edges. The symmetric stress are approximated by nonconforming linear elements and the displacement by piecewise constants. We establish ??(h) error bound in the (broken) L² norm for the divergence of the stress and ??(h) error bound in the L² norm for both the displacement and the stress tensor. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Numerical analysis of a mixed finite element method for a flow-transport problem

We consider a generic flow-transport system of partial differential equations, which has wide application in the waste disposal industry. The approximation of this system, using a finite element method for the brine, radionuclides, and heat combined with a mixed finite element method for the pressure and velocity, is analyzed. Optimal order error estimates in H¹ and L² are derived. The error analysis is given with no restriction on the diffusion tensor. That is, we have included the effects of molecular diffusion and dispersion. © 1996 John Wiley & Sons, Inc.     

Retrieving the time-dependent thermal conductivity of an orthotropic rectangular conductor

The aim of this paper is to determine the thermal properties of an orthotropic planar structure characterized by the thermal conductivity tensor in the coordinate system of the main directions (Oxy) being diagonal. In particular, we consider retrieving the time-dependent thermal conductivity components of an orthotropic rectangular conductor from nonlocal overspecified heat flux conditions. Since only boundary measurements are considered, this inverse formulation belongs to the desirable approach of non-destructive testing of materials. The unique solvability of this inverse coefficient problem is proved based on the Schauder fixed point theorem and the theory of Volterra integral equations of the second kind. Furthermore, the numerical reconstruction based on a nonlinear least-squares minimization is performed using the MATLAB optimization toolbox routine lsqnonlin. Numerical results are presented and discussed in order to illustrate the performance of the inversion for orthotropic parameter identification.     

p-version finite elements and finite cells for finite strain elastoplastic problems

In this paper, the p-version finite element method and its fictitious domain extension, the finite cell method, are extended to the finite strain J₂ plasticity. High-order shape functions are used for the finite element approximation of volume-preserving plastic dominated deformations. The accuracy and efficiency of p-version elements and cells in the finite plastic strain range are evaluated by the computation of two benchmark problems. It is shown that they provide locking free behavior and simplified meshing. These results are verified in comparison with the results of h-version elements in F-bar formulation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local flux mimetic finite difference methods

We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes. To approximate the velocity (vector variable), the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n-gonal mesh face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom per element. A specially chosen quadrature rule for the L ²-product of vector-functions allows for a local flux elimination and reduction of the method to a cell-centered finite difference scheme for the pressure unknowns. Under certain assumptions, first-order convergence is proved for both variables and second-order convergence is proved for the pressure. The assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method. For general polyhedral meshes, non-symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions. Numerical results confirm the theory.     

Cyclic martensitic phase transformation of monocrystals at finite strains

Based on Bain's principle, a C¹-continuous thermo-mechanical micro-macro constitutive model for martensitic phase transformation (PT) of monocrystals at finite strains and hyperelastic free energy function is used. It is represented by a unified non-convex Lagrangian variational functional. The convexification problem is solved here by generalizing the explicit form of the lower Reuss bound for small strains given in [1] to finite strains. Abaqus is used for implementation of 3D finite elements in space, via UMAT-interface which requires Jaumann rate of Kirchhoff stress tensor. Deterministic validation of the model is presented by comparing verified numerical results with experimental data for Cu₈₂Al₁₄Ni₄ [6] for quasiplastic PT. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the differential geometry of frame bundles

Summary The problem to ascertain an admissible structure of frame bundles is solved in this paper, presenting a tensor field H of type-(1.1) which satisfies H³ = H. It acts on the horizontal tensor field as an annihilator and on the vertical tensor field as an almost product structure. When a metric is endowed on the base manifold, it is always possible to assign the metric in the frame bundle such that its element of length obeys the Pythagorean rule when the measurement is done along horizontal and vertical distributions, and by such a general metric it can be proved that the tangent bundle of a frame bundle F(X_n) is reduced to0(n²) ×0(n); especially for n=2m, it is reduced to U(mn) ×0(n). The Lie derivative of H and the parallelism of the three lifts, horizontal, vertical and cnmplete, are examined in terms of their corresponding projection vector fields in the base space.     

Dual-mixed variational formulation and hp finite element method for axisymmetric shell problems in elastodynamics

A new dimensionally reduced axisymmetric shell model is presented briefly for modeling time-dependent problems. This is based on the extended version of the three-field dual-mixed variational formulation of elastostatics [1, 2] to linear elastodynamics, the independent fields of which are the non-symmetric stress tensor, the displacement- and the rotation vector. An important property of the related shell model is that the classical kinematical hypotheses regarding the deformation of the normal to the shell middle surface are not used, i.e., unmodified three-dimensional constitutive equations are applied. The computational performance of the new h- and p-version axisymmetric shell finite elements is tested through a representative cylindrical shell problems. The development presented in this paper has been motivated by the fact that efficient dual-mixed hp plate and shell finite elements were managed previously to be developed for elastostatics by [1-5]. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On $\frak {F}$-normal Fitting classes of finite soluble groups

Let \frak X, \frak F,\frak X\subseteqq \frak F\frak {X}, \frak {F},\frak {X}\subseteqq \frak {F}, be non-trivial Fitting classes of finite soluble groups such that G_{\frak X}G_{\frak {X}} is an \frak X\frak {X}-injector of G for all G ? \frak FG\in \frak {F}. Then \frak X\frak {X} is called \frak F\frak {F}-normal. If \frak F=\frak S_p\frak {F}=\frak {S}_{\pi }, it is known that (1) \frak X\frak {X} is \frak F\frak {F}-normal precisely when \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and consequently (2) \frak F í \frak X\frak N\frak {F}\subseteq \frak {X}\frak {N} implies \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and (3) there is a unique smallest \frak F\frak {F}-normal Fitting class. These assertions are not true in general. We show that there are Fitting classes \frak F\not = \frak S_p\frak {F}\not =\frak {S}_{\pi } filling property (1), whence the classes \frak S_p\frak {S}_{\pi } are not characterized by satisfying (1). Furthermore we prove that (2) holds true for all Fitting classes \frak F\frak {F} satisfying a certain extension property with respect to wreath products although there could be an \frak F\frak {F}-normal Fitting class outside the Lockett section of \frak F\frak {F}. Lastly, we show that for the important cases \frak F=\frak Nⁿ, n\geqq 2\frak {F}=\frak {N}^{n},\ n\geqq 2, and \frak F=\frak S_p1?\frak S_pr, p_i \frak {F}=\frak {S}_{p_{1}}\cdots \frak {S}_{p_{r}},\ p_{i} primes, there is a unique smallest \frak F\frak {F}-normal Fitting class, which we describe explicitly.     

Sparse finite element methods for operator equations with stochastic data

Let A: V → V′ be a strongly elliptic operator on a d-dimensional manifold D (polyhedra or boundaries of polyhedra are also allowed). An operator equation Au = f with stochastic data f is considered. The goal of the computation is the mean field and higher moments of the solution. We discretize the mean field problem using a FEM with hierarchical basis and N degrees of freedom. We present a Monte-Carlo algorithm and a deterministic algorithm for the approximation of the moment for k⩾1. The key tool in both algorithms is a “sparse tensor product” space for the approximation of with O(N(log N) ^k−1) degrees of freedom, instead of N ^k degrees of freedom for the full tensor product FEM space. A sparse Monte-Carlo FEM with M samples (i.e., deterministic solver) is proved to yield approximations to with a work of O(M N(log N) ^k−1) operations. The solutions are shown to converge with the optimal rates with respect to the Finite Element degrees of freedom N and the number M of samples. The deterministic FEM is based on deterministic equations for in D ^k ⊂ ℝ^kd. Their Galerkin approximation using sparse tensor products of the FE spaces in D allows approximation of with O(N(log N) ^k−1) degrees of freedom converging at an optimal rate (up to logs). For nonlocal operators wavelet compression of the operators is used. The linear systems are solved iteratively with multilevel preconditioning. This yields an approximation for with at most O(N (log N) ^k+1) operations. This work was supported under IHP Network “Breaking Complexity” by the Swiss BBW under grant No. 02.0418     

Extension theorems for paraboloids in the finite field setting

In this paper we study the L ^p ? L ^r boundedness of the extension operators associated with paraboloids in ${{\mathbb F}_{q}^{d}}In this paper we study the L ^p − L ^r boundedness of the extension operators associated with paraboloids in \mathbb F_q^d{{\mathbb F}_{q}^{d}} , where \mathbbF_q{\mathbb{F}_{q}} is a finite field of q elements. In even dimensions d ≥ 4, we estimate the number of additive quadruples in the subset E of the paraboloids, that is the number of quadruples (x,y,z,w) ? E⁴{(x,y,z,w) \in E^4} with x + y = z+w. As a result, in higher even dimensions, we obtain the sharp range of exponents p for which the extension operator is bounded, independently of q, from L ^p to L ⁴ in the case when −1 is a square number in \mathbbF_q{\mathbb{F}_{q}} . Using the sharp L ^p −L ⁴ result, we improve upon the range of exponents r, for which the L ² − L ^r estimate holds, obtained by Mockenhaupt and Tao (Duke Math 121:35–74, 2004) in even dimensions d ≥ 4. In addition, assuming that −1 is not a square number in \mathbbF_q{\mathbb{F}_{q}}, we extend their work done in three dimension to specific odd dimensions d ≥ 7. The discrete Fourier analytic machinery and Gauss sum estimates make an important role in the proof.     

Towards dimension expanders over finite fields

In this paper we study the problem of explicitly constructing a dimension expander raised by [3]: Let \mathbbFⁿ \mathbb{F}^n be the n dimensional linear space over the field \mathbbF\mathbb{F}. Find a small (ideally constant) set of linear transformations from \mathbbFⁿ \mathbb{F}^n to itself {A _i } _i∈I such that for every linear subspace V ⊂ \mathbbFⁿ \mathbb{F}^n of dimension dim(V)<n/2 we have

Pointwise supercloseness of tensor‐product block finite elements

In this article, we first introduce interpolation operator of projection type in three dimensions, from which we then derive weak estimates for tensor‐product block finite elements of degree m ≥ 1. Finally, using estimates for the discrete Green's function and the discrete derivative Green's function, we prove that both of the gradient and the function value of the finite element solution u_h and the corresponding interpolant Π_mu of projection type and degree m are superclose in the pointwise sense of the L^∞‐norm. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On skew field coproducts with a finite extension

It is shown that the field coproduct of any skew fieldE with a binomial (commutative) field extensionF/k overk can be expressed as a cyclic extension of a skew fieldK (theE-socle), itself the field coproduct of [F:k] copies ofE overk. Qua vector space the coproduct may also be expressed as a tensor product ofE andK overk. To the memory of Shimshon Amitsur     

Exponential sum estimates over a subgroup in an arbitrary finite field

Let F _q be the finite field consisting of q = p ^r elements and yy an additive character of the field F _q . Take an arbitrary multiplicative subgroup H of size |H| > q ^{C/(log log q)} for some constant C > 0 not largely contained in any multiplicative shift of a subfield. We show that |Σ _h∈H yy(h)| = o(|H|). This means that H is equidistributed in F _q .