Monte Carlo Finite Element Method Research Articles

A multiscale Monte Carlo finite element method for determining mechanical properties of polymer nanocomposites

This paper introduces a multiscale Monte Carlo finite element method (MCFEM) for determining mechanical properties of polymer nanocomposites (PNC) that consist of polymers reinforced with single-walled carbon nanotubes (SWCNT). Note that several approaches discussed in the open literature suggest values for the mechanical properties of PNC that differ significantly from the corresponding ones derived by experimental procedures. This discrepancy is addressed by the proposed MCFEM which accounts for the effect of the non-uniform dispersion and distribution of SWCNT in polymers in the macroscopic mechanical behavior of PNC. Specifically, the method uses a multiscale homogenization approach to link the structural variability at the nano-/micro scales with the local constitutive behavior. Subsequently, the method incorporates a FE scheme to determine the Young’s modulus and Poisson Ratio of PNC. The use of the computed properties in macroscale modeling is validated by comparison with experimental tensile test data.

Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation

This work studies a linear elliptic problem with uncertainty. The introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations. The major emphasis of this work is the probabilistic treatment of uncertainty, addressing the problem of solving linear elliptic boundary value problems with stochastic coefficients. If the stochastic coefficients are known functions of a random vector, then the stochastic elliptic boundary value problem is turned into a parametric deterministic one with solution u(y,x), y∈Γ, x∈D, where D⊂Rd, d=1,2,3, and Γ is a high-dimensional cube. In addition, the function u is specified as the solution of a deterministic variational problem over Γ×D. A tensor product finite element method, of h-version in D and k-, or, p-version in Γ, is proposed for the approximation of u. A priori error estimates are given and an adaptive algorithm is also proposed. Due to the high dimension of Γ, the Monte Carlo finite element method is also studied here. This work compares the asymptotic complexity of the numerical methods, and shows results from numerical experiments. Comments on the uncertainty in the probabilistic characterization of the coefficients in the stochastic formulation are included.

Three-dimensional simulations of the complex dielectric properties of random composites by finite element method

Complex dielectric constants of binary-phase random composites are simulated for a three-dimensional structure consisting of cubic grains using the Monte Carlo-finite element method. Numerical results are fitted using Maxwell–Garnett, Bruggeman symmetrical, and general effective media formulas, and the fitting efficiencies of the formulas are quantitatively evaluated. The general effective media formula gives the best fitting to our simulation results and its accuracy is better than 3.7%. The effects of frequencies on the spatial distribution of electrostatic potentials in dielectric composites are discussed. The distribution of potential contours drawn in the low-frequency region and the high-frequency region show great variation, because of different lengths of time for charge to accumulate near the interphase boundaries. Dielectric spectra are drawn by varying volume fraction and lossy property of one phase in binary-phase composites. General properties of the dielectric spectra are discussed and the characteristics of the dielectric spectra caused by the differences in the lossy properties of the two phases at certain volume fractions are analyzed.

Evaluation of Mixing Rules for Dielectric Constants of Composite Dielectrics by MC-FEM Calculation on 3D Cubic Lattice

Effective dielectric constants of diphase composite dielectrics are simulated by Monte Carlo-finite element method on three-dimensional lattice. Effective dielectric constants with coefficients of variation less than 5.5% are obtained for different ratios of dielectric constants of the two phases, ranging from 10 to 700. Various mixing rules and equations are fitted to these data and the accuracy and relevance of the fits are thoroughly examined. Modified logarithmic rule loses its physical basis when fitted to three-dimensional data. As the ratio of dielectric constants of the two phases increases, the parameters in general Lichterecker mixing rule, general Bruggeman's symmetric equation, general effective media equation and its modified form all increase or decrease monotonously. General effective media equation and its modified form give the best fits to the effective dielectric constants simulated. The simulation results for the dielectric constants of some composite systems are in good agreement with experimental data.

Calculation of the heat deposition and temperature distribution of the target bombarded by high-energy protons using Monte Carlo simulation and finite element method

The Monte Carlo simulation and the finite element methods have been used to calculate the heat deposition and temperature distribution in tungsten plate target when the target is bombarded by high-energy protons from the accelerator with nuclear power of 100 kW. The results show that the heat deposition in the target, reflector and shield will be 48 kW, 15 kW and 11 kW, respectively and the highest temperature in the target plates will be lower than 100°C when the surfaces of plates are cooled by water.

Monte carlo finite element method of structure reliability analysis

Abstract To resolve the problem in reliability of any complicated engineering structures by use of general reliability methodology one often comes across a great deal of limitations due to realistic difficulties. In order to make use of the advantage that the finite element method could numerically resolve the load response of any structure, and the character that Monte Carlo method could carry out stochastic simulation, the Monte Carlo finite element method for any structure reliability analysis was presented. The advantage of high precision, less work in calculation and rapid convergence of the method developed above has been proved by concrete reliability calculation and analysis of the given engineering structure possessing analytic solutions (e.g. a thick wall cylinder). Using the method for rather complicated structures (e.g. high pressure cast steel casing head bearing of oil well head), it was discovered it is necessary to use only a few simulations to obtain the structure reliability value, which took further steps to prove the effectiveness of the Monte Carlo finite element method.