This paper presents the exact integral equation of Hertz's contact problem, which is obtained by taking into account the horizontal displacement, of points in the contacted surfaces due to pressure.

An exact integral expression of finite sum of sequence of a continuously differentiable function has been given, which can be used in approximate calculation with arbitrary accuracy.

By using generalized function, an accurate integration method is developed to get the explicit formulation of the matrixes involved. This can improve the accuracy and efficiency of the method.

Secondly, the exact integration formula of BER for fading channel is presented, moreover, by using the Maclaurin series expansion of MPSK conditional Symbol Error Rate (SER), the simplified BER approximations for BPSK and QPSK are derived over AWGN, Nakagami-m fading and Rayleigh fading channel.

By using an exact integration instead of the approximate integration formula in instantaneous optimal control algorithm,the precise instantaneous optimal control algorithms integration are proposed herein,which includes the precise instantaneous optimal open-loop control,precise instantaneous optimal closed-loop control,and precise instantaneous optimal closed-open-loop control.

According to the scalar diffraction theory,the accurate integral expressions ofthe circular aperture diffraction are derived with the plane wave incidence.

Based on the Mindlin-Reissner plate theory, a new 4-nodes plate bending element with ac-cureately integrated stiffness matrix has been derived through a new isoparametric transformation of a quadrilateral parent element together with the associated shape function.

In this program,the integral on central axis is calculated with accurateformulas of integration,the nonlinearity of the problem is treated by themethod of incremental variable stiffness and the error therein is appropriate-ly corrected.

During the above iteration, the stress resultants of the concrete in the cross section are evaluated by integrating the concrete stress-strain curve over the compression zone, while those of the structural steel and the steel reinforcement (if any) are obtained using the fiber element method.

A method for designing microstructured optical fibers that is based on exact integral equations for the transverse components of the magnetic field of the mode is proposed.

Assuming the loop axis to be aligned with an external static magnetic field, exact integral equations are formulated for the current distribution on the antenna, when it is subject to the excitation of a slice voltage generator.

An exact integral-differential equation is derived for SIT with inhomogeneous broadening, incoherent losses and chirping.

Calculations are also performed using exact integral boundary conditions for the vector potential, as given by the standard electromagnetic field approach, taking into account the effects of both exciting and induced currents.

The exact integral equation of Hertz's contact problem

What's more the quadrilateral element obtained accordingly for the elastic bending of thick plates demonstrates such advantages as high precision for computation and availability of accurate integration for stiffness matrices.

All the strongly singular integrals are computed directly through highly accurate integration techniques.

Accurate integration of phasic signals with frequencies above 0·1 Hz is obtained.

Accurate integration of geostationary orbits with Burdet's focal elements

In this approach several different formulae are applied in a well defined cyclic order to produce highly accurate integration schemes with infinite regions of absolute stability.

Through properly manipulating to the double integration appeared in PO, the exact integration of polygon surfaces could be obtained with independence of frequencies, which is vital for the RCS prediction in the Terahertz band.

In this so called NNCA (NN-Correlation Approximation) the transfer-integral (TI) method allows an exact integration ofΨ2 over the coordinates in all cubic planes but one, leading to the probability-densityP for this plane.

It is difficult to obtain the rigidity matrix by exact integration, and instead, the method of approximate integration is used.

The exact integration have been suggested by various authors for the calculation of rigidity matrix.

This work deals with the exact integration of a Fokker-Planck equation for the mass distribution in heavy ion collisions.

In this paper are presented integral expressions precise enough for computing the internal forces and vertical displacements of double curvature shallow shells under the action of a concentrated load over a certain distance from the boundary according to the moment theory of shells. In case of shallow shells of equal curvatures, these expressions can be easily integrated into formulae which are exactly the same as those given by E. Reissner, but for shallow shells of unequal curvatures, the solution is given...

In this paper are presented integral expressions precise enough for computing the internal forces and vertical displacements of double curvature shallow shells under the action of a concentrated load over a certain distance from the boundary according to the moment theory of shells. In case of shallow shells of equal curvatures, these expressions can be easily integrated into formulae which are exactly the same as those given by E. Reissner, but for shallow shells of unequal curvatures, the solution is given in the form of series composed of Thomson functions, the convergence of which is considered to be quite good in the ordinary scope of application (1 < k1/k2≤2). Furthermore, formulae for the internal forces and vertical displacements of shallow shells of equal curvatures under the action of circular line loads and circular ring loads as well as their exact values at the point of a concentrated load for shallow shells of unequal curvatures are also given in the paper.

Linear form functions are commonly used in a long time for a toroidal volume element swept by a triangle revolved about the symmetrical axis for general axisymmetrical stress problems. It is difficult to obtain the rigidity matrix by exact integration, and as approximations close to the symmetrical axis, the accuracy of this approximation deteriorates very rapidly. The exact integrations have been suggested by some authors for the calculation of rigidity matrix. However, it is shown in this paper that these...

Linear form functions are commonly used in a long time for a toroidal volume element swept by a triangle revolved about the symmetrical axis for general axisymmetrical stress problems. It is difficult to obtain the rigidity matrix by exact integration, and as approximations close to the symmetrical axis, the accuracy of this approximation deteriorates very rapidly. The exact integrations have been suggested by some authors for the calculation of rigidity matrix. However, it is shown in this paper that these exact integrations can only be used for those axisymmetric elastic bodies with central hole. For solid axisymmetric body, it can be proved that the calculation fails due to the divergent property of rigidity matrix integration. In this paper, a new form function is suggested. In this new form function,the radial displacement u vanishes as radial coordinates r approach to zero. The calculated rigidity matrix is convergent everywhere, including these triangular toroidal element closed to the symmetrical axis. This kind of elenent is useful for the calculation of axisymmetric elastic body problem.

In analysis of stress of axisymmetric elastic bodies using finite element method, the mathematical expression for elements of stiffness matrix of triangular elements is a complicated double integration. In order to avoid the complicated calculations the approximate calculation is often preferred in engineering and this results in certain errors inevitably. Beginning with the general integral expression, this paper establishes the error formula in precise and approximate integration and then finds the errors...

In analysis of stress of axisymmetric elastic bodies using finite element method, the mathematical expression for elements of stiffness matrix of triangular elements is a complicated double integration. In order to avoid the complicated calculations the approximate calculation is often preferred in engineering and this results in certain errors inevitably. Beginning with the general integral expression, this paper establishes the error formula in precise and approximate integration and then finds the errors of the elements of stiffness matrix for the element bodies at different distances from the axis of symmetry and for the areas of elements of different sizes. After a large number of numerical evaluations the relation is found between this error and the distenee of the element from the axis of symmetry and the area of element. Approximate formula is given to help the user to choose the right area of element or to see that the divided area of element meets the precision requirements.This paper also compares several methods of approximate calculation for your reference.