A note on integration over finite elements

In the application of the finite element method for structural engineering problems, the evaluation of element stiffness matrices involves the computation of an integral over each element. This note shows that these integrals can be evaluated by a mapping, from the integral over a single element, with the same material properties, in all cases where the sides of the elements are linear in the coordinate system chosen. This brings about a remarkable saving in computation time where numerical integration is used.     

Convergence properties of two nonconforming finite elements

Two nonconforming triangular elements for second-order problems are analyzed. One element is found to be convergent. The other does not yield approximate solutions converging to the solution of the given boundary value problem, but to a wrong limit, which is the solution of another variational equation. This strange convergence behaviour may be typical for a wider class of nonconforming elements violating the basic precondition for convergence. The problem of patch subdivisions is also considered.     

Continuous field approximation of experimentally given data by finite elements

A method for continuous approximation of discretely given data in a two- or three-dimensional domain of arbitrary shape is presented. The method uses shape functions of customary finite elements for modelling the approximation; special properties ensure the smoothness for successful differentiating and improve the accuracy. Many ideas and tools from finite element analysis can be used and the presented examples show the easy application. No restriction is made to the input data, scalar- or vector-valued, and especially the distribution of the given data, can be totally irregular (scattered data). Hence, the method is applicable to a wide range of problems.     

Quantum codes based on fast pauli block transforms in the finite field

Motivated by the fast Pauli block transforms (or matrices) over the finite field GF(q) for an arbitrary number q, we suggest how to construct the simplified quantum code on the basis of quadratic residues. The present quantum code, which is the stabilizer quantum code, can be fast generated from an Abelian group with commutative quantum operators being selected from a suitable Pauli block matrix. This construction does not require the dual-containing or self-orthogonal constraint for the standard quantum error-correction code, thus allowing us to construct a quantum code with much efficiency.     

On the rank of incidence matrices for points and lines of finite affine and projective geometries over a field of four elements

We consider incidence matrices for points and lines of affine and projective geometries over a field of four elements. For such matrices we derive a simple formula for the 2-rank and, as a consequence, new combinatorial identities expressing the relation of the obtained formulas for the rank with previously known formulas. We also present a way to construct generating systems for rows of these matrices.     

一种基于有限域运算的半脆弱数字水印算法

通过分析水印图像置乱加密对数字水印算法的脆弱性影响,提出了一种基于有限域运算的抗JPEG压缩半脆弱数字水印算法。实验证明该算法对固定系数的JPEG压缩有较好的鲁棒性,并对图像的恶意篡改有较好的脆弱性。     

Isomorphic transformations of an elliptic curve over a finite field

Transformations of points of a nonsupersingular elliptic curve are selected as a promising method for the further development of cryptographic systems. Statements are proved on estimates of the number of isomorphic transformations of a nonsupersingular elliptic curve over an extension of a finite field. They vary depending on characteristics of the finite field. The estimates obtained can be used to improve elliptic curve cryptosystems.     

Basis for development of large finite elements locally satisfying all field equations

In a recent paper [1]a modified version of the variational method of Trefftz was presented for plate bending in which the coordinate functions satisfying the nonhomogeneous Lagrange equation were defined on subregions. In [1] the notion of the boundary was generalized to include the interelement boundaries; thus all boundary conditions as well as the interelement continuity requirements were obtained in the form of natural conditions of the generalized boundary. A powerful finite element was developed to illustrate the accuracy of the method.Using a similar concept, two pairs of general variational principles for the solution of problems in the theory of elasticity are now presented and discussed. Making certain assumptions, their application may be shown to yield the lower and upper bounds on the strain energy. Development of some finite elements for applications of practical interest is in progress and will be reported later.     

On the number of nonzero permanents over a finite field of odd characteristic

This note is a comment to [1], where it was proved that, over a field of odd characteristic, the number of square matrices of order n, n ≥ 3, with nonzero permanent is greater than the number of square matrices of the same order with nonzero determinant.     

A method for computation of the discrete Fourier transform over a finite field

The discrete Fourier transform over a finite field finds applications in algebraic coding theory. The proposed computation method for the discrete Fourier transform is based on factorizing the transform matrix into a product of a binary block circulant matrix and a diagonal block circulant matrix.     

利用有限域上向量空间构造Cartesian认证码

利用有限域上的向量空间构造了一个新的Cartesian认证码,并计算了该认证码的参数。当编码规则按等概率分布选取时,求出了敌方模仿攻击成功的概率PI和替换攻击成功的概率PS,进而得到此认证码为一个PI最优的Cartesian认证码。     

Unsteady boundary layer over an impulsively started flat plate using finite elements

The problem under consideration is that of a suddenly accelerated semi-infinite flat plate in an incompressible viscous fluid. The present analysis is based upon the use of finite elements to idealize the flow over the plate and solving the nonlinear unsteady boundary layer equation by Galerkin's procedure and the results are compared with those obtained previously using a mixed explicit-implicit finite difference scheme. An average number of three iterations are needed to obtain a solution within a maximum tolerance between interates less than 0.01%. The method has successfully demonstrated the exponential decay of the velocity at large times, which has been reported by previous investigators. The transition region of the flow is terminated to reach Blasius state, when the plate has moved a distance equal to six times the distance from the station into consideration to the leading edge of the plate. The methods shows excellent accuracy in dealing with the nonlinear equation of the unsteady viscous flow.     

The simulation of texture evolution with finite elements over orientation space I. Development

A new class of finite element schemes is presented for the modeling of arbitrary polycrystalline microstructure. A framework is constructed by viewing texture as arising out of maps that associate crystals with orientations drawn from a fundamental region of orientation space. The state of the microstructure is characterized by an orientation distribution function and a hardness field over a fundamental region. Differential equations are derived to describe the evolution of the two fields. Finite element schemes, cast over discretized fundamental regions, are considered for the solution of the differential set. Properties of the schemes considered are illustrated by application to the texturing of a planar polycrystalline microstructure.     

Neural classification of finite elements

The work deals with a comparative performance of finite elements, making use of their formulation as vectors (or patterns) in a multi-dimensional space of proper attributes. Since the attributes control the performance, elements defined by similar patterns and related to the same class should show similar behavior. The pattern classification may be carried out with the help a self-organizing feature map of Kohonen with the patterns corresponding to the input space. These networks learn both the distribution and topology of a set of input space. At the end of the learning process, the neurons become selectively tuned to classes of input patterns, thus specifying “family relationships” among the elements. The work makes use of the four attributes: the element dimensionality, its number of nodes, maximum degree of interpolation polynomials and number of degrees of freedom per node, though a more general characterization is also possible.     

有限素域上椭圆曲线模逆运算的设计与实现

在对四种不同类型的求模逆算法进行改进的基础上,提出了一种统一的有限素数域上的模逆运算结构。该结构结合这四种类型的模逆算法,通过选择信号完成Montgomery模逆或一般整数模逆运算,而不增加其它的硬件资源消耗。最后对该结构采用VHDL硬件描述语言进行了代码设计,并基于FPGA进行了编译综合和布局布线。实验结果表明该设计与采用两种不同结构分别计算的方案相比,节省近一半的硬件资源。     

Minimal-degree thin triangular plate and shell bending finite elements of order two and four

Triangular bending finite elements for the analysis of thin plates are created with shear. Only the minimal number of shear degrees of freedom needed to maintain interelement continuity for a given element order are retained. The result is a 9-dof element of order 2, and a 12-dof element of order 4. A 15-dof element of order 6 is impossible.     

On the Berlekamp/Massey algorithm and counting singular Hankel matrices over a finite field

We derive an explicit count for the number of singular n×n$n\times n$ Hankel (Toeplitz) matrices whose entries range over a finite field with q$q$ elements by observing the execution of the Berlekamp/Massey algorithm on its elements. Our method yields explicit counts also when some entries above or on the anti-diagonal (diagonal) are fixed. For example, the number of singular n×n$n\times n$ Toeplitz matrices with 0’s on the diagonal is q²ⁿ⁻³+qⁿ⁻¹−qⁿ⁻²$q^{2n-3}+q^{n-1}-q^{n-2}$.     

On the development of traveling load finite elements

Specialized finite elements are devised to handle traveling load problems in stationary and moving/rotating semi-infinite and body of revolution structures (slabs, rollers, tires, etc.). The generality of the elements is such that the full range of rectilinear/rotary loading and structural velocities can be handled. Hence, such phenomena as resonances and standing as well as traveling sub/supersonic waves can be simulated. The accuracy and capabilities of the elements is demonstrated by several numerical examples.