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When faced with the mathematical modeling of any engineering system, whether for design, performance assessment, optimization, or control, the engineer has to determine the level of accuracy versus the simplicity of the mathematical formulation. Although it is universally accepted that the more complex the formulation, the more accurate the results will be, these usually come at the expense of larger CPU times, a substantial amount of computer resources, and are generally limited by the capacity of computers and computing power, which most times precludes their use in favor of simpler models. Many times, however, engineers do not realize the potential risk that oversimplification of a problem generates in terms of accuracy of the results; that is, the model solution does not resemble the system behavior. Through a demonstrative example, the present study addresses the issue of oversimplification of the resulting mathematical model and the corresponding accuracy of its solution. After constructing three sets of models of the physical system, each with a different level of detail, the solutions are then compared to experimental data. The results show that the accuracy of the numerical approximation depends directly on the level of complexity of the mathematical model used, and that oversimplification may result in up to a ninefold degradation of the results. In addition, minor changes in the inlet boundary condition and geometry result in significant changes in the flow pattern, up to a fivefold difference between different models in the recirculation bubble relative error. This information is fundamental for engineering professionals to consider during the modeling process in applications. [ABSTRACT FROM AUTHOR]

Many mathematical and physical problems are led to find a root of a real functionf. This kind of equation is an inverse problem and it is difficult to solve it. Especially in engineering sciences, the analytical expression of the functionfis unknown to the experimenter, but it can be measured at each pointxkwithM(xk) as expected value and induced error ξk. The aim is to approximate the unique root θ under some assumptions on the functionfand errors ξk. We use a stochastic approximation algorithm that constructs a sequence (xk)k ⩾ 1. We establish the almost complete convergence of the sequence (xk)kto the exact root θ by considering the errors (ξk)kquasi-associated and we illustrate the method by numerical examples to show its efficiency. [ABSTRACT FROM AUTHOR]

In this study, we evaluated computational efficiency of finite element (FE) simulations when a numerical approximation method was used to obtain the tangent moduli. A fiber-reinforced hyperelastic material model for nearly incompressible soft tissues was implemented for 3D solid elements using both the approximation method and the closed-form analytical method, and validated by comparing the components of the tangent modulus tensor (also referred to as the material Jacobian) between the two methods. The computational efficiency of the approximation method was evaluated with different perturbation parameters and approximation schemes, and quantified by the number of iteration steps and CPU time required to complete these simulations. From the simulation results, it can be seen that the overall accuracy of the approximation method is improved by adopting the central difference approximation scheme compared to the forward Euler approximation scheme. For small-scale simulations with about 10,000 DOFs, the approximation schemes could reduce the CPU time substantially compared to the closed-form solution, due to the fact that fewer calculation steps are needed at each integration point. However, for a large-scale simulation with about 300,000 DOFs, the advantages of the approximation schemes diminish because the factorization of the stiffness matrix will dominate the solution time. Overall, as it is material model independent, the approximation method simplifies the FE implementation of a complex constitutive model with comparable accuracy and computational efficiency to the closed-form solution, which makes it attractive in FE simulations with complex material models. [ABSTRACT FROM AUTHOR]

Variational image restoration models for both additive and multiplicative noise (MN) removal are rarely encountered in the literature. This paper proposes a new variational model and a fast algorithm for its numerical approximation to remove independent additive and MN from digital images. Two previous works by L. Rudin, S. Osher, and E. Fatemi [Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992), pp. 259–268] and Z. Jin and X. Yang [Analysis of a new variational model for multiplicative noise removal, J. Math. Anal. Appl. 362 (2010), pp. 415–426] are used to develop the new model. As a result, developing a fast numerical algorithm is difficult because the associated Euler–Lagrange equation is highly nonlinear and standard unilevel iterative methods are not appropriate. To this end, we develop an efficient nonlinear multigrid approach via a robust fixed-point smoother. Numerical tests using both synthetic and realistic images not only confirm that our new model delivers quality results but also that the proposed numerical algorithm allows a very fast numerical realization of the model. [ABSTRACT FROM PUBLISHER]

A new sparse kernel probability density function (pdf) estimator based on zero-norm constraint is constructed using the classical Parzen window (PW) estimate as the target function. The so-called zero-norm of the parameters is used in order to achieve enhanced model sparsity, and it is suggested to minimize an approximate function of the zero-norm. It is shown that under certain condition, the kernel weights of the proposed pdf estimator based on the zero-norm approximation can be updated using the multiplicative nonnegative quadratic programming algorithm. Numerical examples are employed to demonstrate the efficacy of the proposed approach. [ABSTRACT FROM PUBLISHER]

There is a significant body of literature wherein a linear approximation of Escaig's model is used to justify the large experimentally measured activation-volumes for cross-slip in face-centered-cubic copper. Here, by examining the error between the linear approximation and the original theory, we show that this explanation is not satisfactory. The calculated value for activation volume in copper, using Escaig's original equations, yields ∼60b3 (b = Burgers vector) while the linear approximation yields 200b3, the latter result fortuitously matching the experimental values. [ABSTRACT FROM AUTHOR]

We investigate an application of the method of fundamental solutions (MFS) to the one-dimensional parabolic inverse Cauchy-Stefan problem, where boundary data and the initial condition are to be determined from the Cauchy data prescribed on a given moving interface. In [B.T. Johansson, D. Lesnic, and T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan Problem, Appl. Math Model. 35 (2011), pp. 4367-4378], the inverse Stefan problem was considered, where only the boundary data is to be reconstructed on the fixed boundary. We extend the MFS proposed in Johansson et al. (2011) and show that the initial condition can also be simultaneously recovered, i.e. the MFS is appropriate for the inverse Cauchy-Stefan problem. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be efficiently obtained with small computational cost. [ABSTRACT FROM AUTHOR]