Interval finite element method and its application on anti-slide stability analysis

The problem of interval correlation results in interval extension is discussed by the relationship of interval-valued functions and real-valued functions. The methods of reducing interval extension are given. Based on the ideas of the paper, the formulas of sub-interval perturbed finite element method based on the elements are given. The sub-interval amount is discussed and the approximate computation formula is given. At the same time, the computational precision is discussed and some measures of improving computational efficiency are given. Finally, based on sub-interval perturbed finite element method and anti-slide stability analysis method, the formula for computing the bounds of stability factor is given. It provides a basis for estimating and evaluating reasonably anti-slide stability of structures.

INTERVAL ARITHMETIC AND STATIC INTERVAL FINITE ELEMENT METHOD

When the uncertainties of structures may be bounded in intervals, through some suitable discretization, interval finite element method can be constructed by combining the interval analysis with the traditional finite element method (FEM). The two parameters, median and deviation, were used to represent the uncertainties of interval variables. Based on the arithmetic rules of intervals, some properties and arithmetic rules of interval variables were demonstrated. Combining the procedure of interval analysis with FEM, a static linear interval finite element method was presented to solve the non-random uncertain structures. ne solving of the characteristic parameters of n-freedom uncertain displacement field of the static governing equation was transformed into 2 n-order linear equations. It is shown by a numerical example that the proposed method is practical and effective.

Interval finite difference method for steady-state temperature field prediction with interval parameters

A new numerical technique named interval finite difference method is proposed for the steady-state temperature field prediction with uncertainties in both physical parameters and boundary conditions. Interval variables are used to quantitatively describe the uncertain parameters with limited information. Based on different Taylor and Neumann series, two kinds of parameter perturbation methods are presented to approximately yield the ranges of the uncertain temperature field. By comparing the results with traditional Monte Carlo simulation, a numerical example is given to demonstrate the feasibility and effectiveness of the proposed method for solving steady-state heat conduction problem with uncertain-but-bounded parameters.

Equivalent method for accurate solution to linear interval equations

Based on linear interval equations, an accurate interval finite element method for solving structural static problems with uncertain parameters in terms of optimization is discussed. On the premise of ensuring the consistency of solution sets, the original interval equations are equivalently transformed into some deterministic inequations. On this basis, calculating the structural displacement response with interval parameters is predigested to a number of deterministic linear optimization problems. The results are proved to be accurate to the interval governing equations. Finally, a numerical example is given to demonstrate the feasibility and efficiency of the proposed method.

FUZZY ARITHMETIC AND SOLVING OF THE STATIC GOVERNING EQUATIONS OF FUZZY FINITE ELEMENT METHOD

The key component of finite element analysis of structures with fuzzy parameters, which is associated with handling of some fuzzy information and arithmetic relation of fuzzy variables, was the solving of the governing equations of fuzzy finite element method. Based on a given interval representation of fuzzy numbers, some arithmetic rules of fuzzy numbers and fuzzy variables were developed in terms of the properties of interval arithmetic. According to the rules and by the theory of interval finite element method, procedures for solving the static governing equations of fuzzy finite element method of structures were presented. By the proposed procedure, the possibility distributions of responses of fuzzy structures can be generated in terms of the membership functions of the input fuzzy numbers. It is shown by a numerical example that the computational burden of the presented procedures is low and easy to implement. The effectiveness and usefulness of the presented procedures are also illustrated.

Flexural wave bandgap properties of phononic crystal beams with interval parameters

Uncertainties are unavoidable in practical engineering,and phononic crystals are no exception.In this paper,the uncertainties are treated as the interval parameters,and an interval phononic crystal beam model is established.A perturbation-based interval finite element method(P-IFEM)and an affine-based interval finite element method(A-IFEM)are proposed to study the dynamic response of this interval phononic crystal beam,based on which an interval vibration transmission analysis can be easily implemented and the safe bandgap can be defined.Finally,two numerical examples are investigated to demonstrate the effectiveness and accuracy of the P-IFEM and A-IFEM.Results show that the safe bandgap range may even decrease by 10%compared with the deterministic bandgap without considering the uncertainties.

An interval finite element method based on bilevel Kriging model

This study introduces a new approach utilizing an interval finite element method combined with a bilevel Kriging model to determine the bounds of structural responses in the presence of spatial uncertainties.A notable benefit of this approach is its ability to determine the response bounds across all degrees of freedom with a small sample size,which means that it has high efficiency.Firstly,the spatially varying uncertain parameters are quantified using an interval field model,which is described by a series of standard interval variables within a truncated interval Karhunen-Loe`ve(K-L)series expansion.Secondly,considering that the bound of structural response is a function of spatial position with the property of continuity,a surrogate model for the response bound is constructed,namely the first-level Kriging model.The training samples required for this surrogate model are obtained by establishing the second-level Kriging model.The second-level Kriging model is established to describe the structural responses at particular locations relative to the interval variables so as to facilitate the upper and lower bounds of the node response required by the first-level Kriging model.Finally,the accuracy and effectiveness of the method are verified through examples.

A RIEMANN-HILBERT APPROACH TO THE INITIAL-BOUNDARY PROBLEM FOR DERIVATIVE NONLINEAR SCHRDINGER EQUATION

We use the Fokas method to analyze the derivative nonlinear Schrodinger （DNLS） equation iqt （x, t） = -qxx （x, t）＋（rq^2）x on the interval [0, L]. Assuming that the solution q（x, t） exists, we show that it can be represented in terms of the solution of a matrix Riemann- Hilbert problem formulated in the plane of the complex spectral parameter ξ. This problem has explicit （x, t） dependence, and it has jumps across {ξ∈C｜Imξ^4 = 0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a（ξ）, b（ξ）}, {A（ξ）, B（ξ）}, and {A（ξ）, B（ξ）}, which in turn are defined in terms of the initial data q0（x） = q（x, 0）, the bound- ary data g0（t）= q（0, t）, g1（t） = qx（0, t）, and another boundary values f0（t） = q（L, t）, f1（t） = qx（L, t）. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.

Influence Analysis of Machining and Installation Errors on the Radial Stiffness of a Non-Pneumatic Mechanical Elastic Wheel

Machining and installation errors are unavoidable in mechanical structures. However, the effect of errors on radial stiffness of the mechanical elastic wheel(ME-Wheel) is not considered in previous studies. To this end, the interval mathematical model and interval finite element model of the ME-Wheel were both established and compared with bench test results. The intercomparison of the influence of the machining and installation errors on the ME-Wheel radial stiffness revealed good consistency among the interval mathematical analysis, interval finite element simulation,and bench test results. Within the interval range of the ME-Wheel machining and installation errors, parametric analysis of the combined elastic rings was performed at different initial radial rigidity values. The results showed that the initial radial stiffness of the flexible tire body significantly influenced the ME-Wheel radial stiffness, and the inverse relationship between the hinge unit length or suspension hub and the radial stiffness was nonlinear. The radial stiffness of the ME-Wheel is predicted by using the interval algorithm for the first time, and the regularity of the radial stiffness between the error and the load on the ME-Wheel is studied, which will lay the foundation for the exact study of the ME-Wheel dynamic characteristics in the future.

An interval finite element method for electromagnetic problems with spatially uncertain parameters

During the manufacturing process of dielectric materials used in electromagnetic engineering, the electromagnetic parameters are often spatially uncertain due to the processing technology, environmental temperature, personal operations, etc. Traditionally,the random field model can be used to measure the spatial uncertainties, but its construction requires a large number of samples.On the contrary, the interval field model only needs the upper and lower bounds of the spatially uncertain parameters, which requires much less samples and furthermore is easy to understand and use for engineers. Therefore, in this paper, the interval field model is introduced to describe the spatial uncertainties of dielectric materials, and then an interval finite element method(IFEM) is proposed to calculate the upper and lower bounds of electromagnetic responses. Firstly, the interval field of the dielectric material is represented by the interval K-L expansion and inserted into the scalar Helmholtz wave equations, and thus the interval equilibrium equations are constructed according to the node-based finite element method. Secondly, a perturbation interval finite element method is developed for calculating the upper and lower bounds of electromagnetic responses such as the electric strength and magnetic strength. Finally, the effectiveness of the proposed method is verified by three numerical examples.

INTERVAL FINITE VOLUME METHOD FOR UNCERTAINTY SIMULATION OF TWO-DIMENSIONAL RIVER WATER QUALITY

Under the interval uncertainties, by incorporating the discretization form offinite volume method and interval algebra theory, an Interval Finite Volume Method (IF-VM) wasdeveloped to solve water quality simulation issues for two-dimensional river when lacking effectivedata of flow velocity and flow quantity. The IFVM was practically applied to a segment of theXiangjiang River because the Project of Hunan Inland Waterway Multipurpose must be started workingafter the environmental impact assessment for it. The simulation results suggest that there existrather apparent pollution zones of BOD_5 downstream the Dongqiaogang discharger and that of CODdownstream Xiaoxiangjie discharger, but the pollution sources have no impact on the safety of thethree water plants located in this river segment. Although the developed IFVM is to be perfected, itis still a powerful tool under interval uncertainties for water environmental impact assessment,risk analysis, and water quality planning, etc. besides water quality simulation studied in thispaper.

Uncertain eigenvalue analysis of the dielectric-filled waveguide by an interval vector finite element method

Eigenvalues of the dielectric-filled waveguide are of great importance to its transmission characteristic analysis and optimization design, which could be easily affected by spatially uncertain dielectric parameters. For the sake of quantifying their influence on eigenvalues of the dielectric-filled waveguide and overcoming the limitation of less samples, an interval vector finite element method(IVFEM) is proposed to acquire the lower and upper bounds of the eigenvalues with spatial uncertainty of the medium parameters. Firstly, the uncertain dielectric material properties are described by the interval field model and the corresponding interval Karhunen-Loève(K-L) approximate method. Secondly, by inserting the interval uncertainties into the constitutive relationship of the standard generalized eigenvalue equations of the dielectric-filled waveguide, an interval standard generalized eigenvalue equation is then formulated. At last, the lower and upper bounds of the eigenvalues are calculated according to the first-order perturbation method, which can be used to estimate the transmission properties of the waveguide efficiently. Three kinds of the dielectric-filled waveguides are analyzed by the proposed IVFEM and verified by Monte Carlo simulation method.

On the Number of Finite Non-isomorphic Abelian Groups in Short Intervals

Let the arithmetic function a(n) denote the number of non-isomorphic Abeliangroups of order n;k, positive integer, and x≥0. We setA_k(x)= sum from n≤x a(n)=k to (1)andA_k(x;h) =A_k(x+h)-A_k(x). A. Ivice first investigated the distribution of the values of finite non-isomorphicAbelian groups in short intervals. E. Kratzel reduced the problem to estimate theerror term △(1, 2, 3;x) in the three-dimensional multiplicative problem, and furtherimproved Ivice’s result.