Nonrecursive residual Monte Carlo method for S_(N) transport discretization error estimation

In this paper,we present a nonrecursive residual Monte Carlo method for estimating discretization errors associated with the S_(N) transport solution to radiation transport problems.Although this technique is general,we applied it to the mono-energetic 1-D S_(N) equation with linear-discontinuous finite element method spatial discretization as a demonstration of the theory for the purpose of this study.Two angular flux representations:conforming and simplified representations were considered in this analysis,and the results were compared.It is shown that the simplified representation dramatically reduces the memory footprint and computational complexity of residual source generation and sampling while accurately capturing the error associated with certain types of responses.

Accuracy analysis of immersed boundary method using method of manufactured solutions

The immersed boundary method is an effective technique for modeling and simulating fluid-structure interactions especially in the area of biomechanics.This paper analyzes the accuracy of the immersed boundary method.The procedure contains two parts,i.e.,the code verification and the accuracy analysis.The code verification provides the confidence that the code used is free of mistakes,and the accuracy analysis gives the order of accuracy of the immersed boundary method.The method of manufactured solutions is taken as a means for both parts.In the first part,the numerical code employs a second-order discretization scheme,i.e.,it has second-order accuracy in theory.It matches the calculated order of accuracy obtained in the numerical calculation for all variables.This means that the code contains no mistake,which is a premise of the subsequent work.The second part introduces a jump in the manufactured solution for the pressure and adds the corresponding singular forcing terms in the momentum equations.By analyzing the discretization errors,the accuracy of the immersed boundary method is proven to be first order even though the discretization scheme is second order.It has been found that the coarser mesh may not be sensitive enough to capture the influence of the immersed boundary,and the refinement on the Lagrangian markers barely has any effect on the numerical calculation.

Uncertainty Propagation of Structural Computation for Fatigue Assessment

Offshore wind substations are subjected to uncertain loads from waves,wind and currents.Sea states are composed of irregular waves which statistics are usually characterized.Irregular loads may induce fatigue failure of some structural components of the structures.By combining fatigue damage computed through numerical simulations for each sea state endured by the structure,it is possible to assess fatigue failure of the structure over the whole deployment duration.Yet,the influence of the discretization error on the fatigue damage is rarely addressed.It is possible to estimate the discretization error on the quantity of interest computed at the structural detail suspected to fail.However,the relation between this local quantity of interest and the fatigue damage is complex.In this paper,a method that allows propagating error bounds towards fatigue damage is proposed.While increasing computational burden,computing discretization error bounds is a useful output of finite element analysis.It can be utilized to either validate mesh choice or guide remeshing in case where potential error on the fatigue damage is too large.This method is applied to an offshore wind substation developped by Chantiers de l’Atlantique using two discretization error estimators in a single sea state.

ERROR ESTIMATES FOR THE TIME DISCRETIZATION FOR NONLINEAR MAXWELL'S EQUATIONS

This paper is devoted to the study of a nonlinear evolution eddy current model of the type δtB（H） ＋ △↓× （ △↓ × H） =0 subject to homogeneous Dirichlet boundary conditions H × v = 0 and a given initial datum. Here, the magnetic properties of a soft ferromagnet are linked by a nonlinear material law described by B（H）. We apply the backward Euler method for the time discretization and we derive the error estimates in suitable function spaces. The results depend on the nonlinearity of B（H）.

Hyperbolic Conservation Laws on Manifolds.An Error Estimate for Finite Volume Schemes

Following Ben-Artzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L1-error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. The error in the L1 norm is of order h1/4 at most, where h represents the maximal diameter of elements in the family of geodesic triangulations. The proof relies on a suitable generalization of Cockburn, Coquel, and LeFloch＇s theory which was originally developed in the Euclidian setting. We extend the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties.

Computational uncertainty principle in nonlinear ordinary differential equations——Ⅱ.Theoretical analysis

The error propagation for general numerical method in ordinarydifferential equations ODEs is studied. Three kinds of convergence, theoretical, numerical and actual convergences, are presented. The various components of round-off error occurring in floating-point computation are fully detailed. By introducing a new kind of recurrent inequality, the classical error bounds for linear multistep methods are essentially improved, and joining probabilistic theory the “normal” growth of accumulated round-off error is derived. Moreover, a unified estimate for the total error of general method is given. On the basis of these results, we rationally interpret the various phenomena found in the numerical experiments in part I of this paper and derive two universal relations which are independent of types of ODEs, initial values and numerical schemes and are consistent with the numerical results. Furthermore, we give the explicitly mathematical expression of the computational uncertainty principle and expound the intrinsic relation between two uncertainties which result from the inaccuracies of numerical method and calculating machine.