In this paper, we study the role of mesh quality on the accuracy of linear finite element approximation. We derive a more detailed error estimate, which shows explicitly how the shape and size of elements, and symmetry structure of mesh effect on the error of numerical approximation. Two computable parameters Ge and Gv are given to depict the cell geometry property and symmetry structure of the mesh. In compare with the standard a priori error estimates, which only yield information on the asymptotic error behaviour in a global sense, our proposed error estimate considers the effect of local element geometry properties, and is thus more accurate. Under certain conditions, the traditional error estimates and supercovergence results can be derived from the proposed error estimate. Moreover, the estimators Ge and Gv are computable and thus can be used for predicting the variation of errors. Numerical tests are presented to illustrate the performance of the proposed parameters Ge and Gv.

The Jiles-Atherton (J-A) model is a commonly used physics-based model in describing the hysteresis characteristics of ferromagnetic materials. However, citations and interpretation of this model in literature have been non-uniform. Solution methods for solving numerically this model has not been studied adequately. In this paper, through analyzing the mathematical properties of equations and the physical mechanism of energy conservation, we point out some unreasonable descriptions of this model and develop a relatively more accurate, modified J-A model together with its numerical solution method. Our method employs a fixed point method to compute anhysteretic magnetization. We obtain the susceptibility value of the anhysteretic magnetization analytically and apply the 4th order Runge-Kutta method to the solution of total magnetization. Computational errors are estimated and then precisions of the solving method in describing various materials are verified. At last, through analyzing the effects of the accelerating method, iterative error and step size on the computational errors, we optimize the numerical method to achieve the effects of high precision and short computing time. From analysis, we determine the range of best values of some key parameters for fast and accurate computation.

In the paper, we present an efficient two-grid method for the approximation of two-dimensional nonlinear reaction-diffusion equations using a expanded mixed finite-element method. We transfer the nonlinear reaction diffusion equation into first order nonlinear equations. The solution of the nonlinear system on the fine space is reduced to the solutions of two small (one linear and one non-linear) systems on the coarse space and a linear system on the fine space. Moreover, we obtain the error estimation for the two-grid algorithm. It is showed that coarse space can be extremely coarse and achieve asymptotically optimal approximation as long as the mesh sizes satisfy . An numerical example is also given to illustrate the effectiveness of the algorithm.

The radiative transfer equation (RTE) arises in many different areas of science and engineering. In this paper, we propose and investigate a discrete-ordinate discontinuous-streamline diffusion (DODSD) method for solving the RTE, which is a combination of the discrete-ordinate technique and the discontinuous-streamline diffusion method. Different from the discrete-ordinate discontinuous Galerkin (DODG) method for the RTE, an artificial diffusion parameter is added to the test functions in the spatial discretization. Stability and error estimates in certain norms are proved. Numerical results show that the proposed method can lead to a more accurate approximation in comparison with the DODG method.

This study aimed at developing allometric models from destructive sample field data for estimating both aboveground and belowground tree biomass and assessing changes in root biomass after old-growth Brachystegia-Julbernardia (miombo) woodland clearing in central Zambia. Logarithmic linear models were selected for estimating tree biomass because they gave the most accurate (low mean error) predictions. On average aboveground and belowground biomass in regrowth woodland represented 29% and 41%, respectively, of the biomass in old-growth woodland. The root:shoot ratios were 0.54 and 0.77 in old-growth and regrowth woodland, respectively. Ten years after clear-cutting old-growth woodland, root biomass loss was about 60% of the original biomass. The main cause of post clearing root biomass loss was fire which at the study sites occurred annually or biannually. Control of fire in cleared sites should be encouraged in forest management for carbon storage and sequestration in miombo woodland of southern Africa.

We consider the following problem of error estimation for the optimal control ofnonlinear parabolic partial differential equations: let an arbitrary admissible controlfunction be given. How far is it from the next locally optimal control? Under naturalassumptions including a second-order sufficient optimality condition for the (unknown)locally optimal control, we estimate the distance between the two controls. To do this, weneed some information on the lowest eigenvalue of the reduced Hessian. We apply thistechnique to a model reduced optimal control problem obtained by proper orthogonaldecomposition (POD). The distance between a local solution of the reduced problem to alocal solution of the original problem is estimated.

The unsteady motion of a fully immersed solid sphere at low to moderate particle Reynolds number, from 1 to about 1600, has been modeled by considering the quasi-steady viscous drag, the added mass force, and the history force. The last force results from unsteady boundary layer growth and is often neglected in multiphase flow applications due to its complex formula. However, it has been shown that this unsteady viscous force is more crucial than the widely employed added mass force to describe an unsteady sphere motion. Thus, targeting the flow regime when the quasi-steady viscous drag is dominating, this work proposes two simple formulae to approximate the errors of neglecting either the history force or the added mass force in describing a fully immersed spherical pendulum motion. The proposed formulae are shown to capture the actual error when the numerical solution of a partial model that omits one force component is compared to the full model prediction when the sphere density is not too close to the ambient liquid.

Germination tests are performed on a routine basis to determine the viability of genebank accessions. The results determine which accessions have to be rejuvenated. The reliability of the germination test results used by the Centre for Genetic Resources, the Netherlands was determined by the retesting of 641 random samples anonymously, in the same year and by the same testing agency as the original tests. Results showed alarmingly low reliabilities, with error levels much higher than expected based on sampling effects. The result of a germination test of a random sample with a germination of 80% was shown to have a 95% confidence interval from 63 to 97%. The errors differed strongly over crops and testing years, and were larger for crop wild relatives than for crop species.

The present paper reveals an analytically computational method for the inverse Cauchy problem of Laplace equation. For the sake of analyticity, and also for the frequent use of rectangular plate in engineering structure, we only consider the analytical solution in a two-dimensional rectangular domain, wherein a missing boundary condition is recovered from a partial measurement of the Neumann data on an accessible boundary. The Fourier series is used to formulate a first-kind Fredholm integral equation for the unknown function of data. Then, we consider a Lavrentiev regularization amended to a second-kind Fredholm integral equation. The termwise separable property of kernel function allows us to obtain a closed-form solution of the regularization type. The uniform convergence and error estimation of the regularization solution are proven. The numerical examples show the effectiveness and robustness of the new method.

A benchmark is organised to quantify the variability relative to structure dynamicscomputations. The chosen demonstrator is a pump in service in thermal central units, whichis an engineered system with not well-known parameters, considered in its workenvironment. The blind modal characterisation of the separate pump components shows a5%–12% variability on eigenfrequency values and a less than 15% frequency error incomparison with experimental values. The numerical-experimental MAC numbers reach 0.7 atthe maximum, even after updating. An example of modal results on the pump assembly fixedis presented, which shows a larger discrepancy with measurement values, essentially due tothe modelling of the interfaces and boundary condition, and to the possible simplificationof the main components F.E. models to reduce their size. Though a significant frequencyerror, the first overall modes are correctly identified. If this tendency can be confirmedfrom all the participants’ results, the conclusion to be drawn is that, if the predictivecapability of F.E. models to represent the dynamical behaviour of sub-structures issatisfactory, the one relative to structures that are built-up of several components doesnot allow their confident use. Additional information issued from measurements is neededto improve their accuracy.

In this paper we construct a new H(div)-conforming projection-based p-interpolation operator that assumes only Hr(K) $\cap$${\bf \tilde H}$-1/2(div, K)-regularity (r > 0) on the reference element (either triangle or square) K. We show that this operator is stable with respect to polynomial degrees and satisfies the commuting diagram property. We also establish an estimate for the interpolation error in the norm of the space ${\bf \tilde H}$-1/2(div, K), which is closely related to the energy spaces for boundary integral formulations of time-harmonic problems of electromagnetics in three dimensions.

Despite its known effectiveness, a typical vibratory assembly method tends to generate adverse impact forces between mating parts commensurate with the relatively large vibratory motion required for reliably compensating positioning errors of arbitrary magnitude. To this end, this paper presents a neural network-based vibratory assembly method with its emphasis on reducing the mating forces for chamferless prismatic parts. In this method, the interactive force is effectively suppressed by reducing the amplitude of vibratory motion, while the greater part of the relative positioning error is estimated and compensated by a neural network. The estimation performance of the neural network and the overall performance of the assembly method are evaluated experimentally. Experimental results show that the assembly is efficiently accomplished with small reaction forces, and the possible insertion error range is also expanded