Physically Consistent Scar Tissue Dynamics from Scattered Set of Data: A Novel Computational Approach to Avoid the Onset of the Runge Phenomenon

The foreign body reaction is a complex biological process leading to the insulation of implanted artificial materials through a capsule of scar tissue. In particular, in chronic implantations of neural electrodes, the prediction of the scar tissue evolution is crucial to assess the implant reliability over time. Indeed, the capsule behaves like an increasing insulating barrier between electrodes and nerve fibers. However, no explicit and physically based rules are available to computationally reproduce the capsule evolution. In addition, standard approaches to this problem (i.e., Vandermonde-based and Lagrange interpolation) fail for the onset of the Runge phenomenon. More specifically, numerical oscillations arise, thus standard procedures are only able to reproduce experimental detections while they result in non physical values for inter-interval times (i.e., times before and after experimental detections). As a consequence, in this work, a novel framework is described to model the evolution of the scar tissue thickness, avoiding the onset of the Runge phenomenon. This approach is able to provide novel approximating functions correctly reproducing experimental data (R2≃0.92) and effectively predicting inter-interval detections. In this way, the overall performances of previous approaches, based on phenomenological fitting polynomials of low degree, are improved.

A new approach for constructing mock-Chebyshev grids

Polynomial interpolation with equidistant nodes is notoriously unreliable due to the Runge phenomenon, and is also numerically ill-conditioned. By taking advantage of the optimality of the interpolation processes on Chebyshev nodes, one of the best strategies to defeat the Runge phenomenon is to use the mock-Chebyshev points, which are selected from a satisfactory uniform grid, for polynomial interpolation. Yet, little literature exists on the computation of these points. In this study, we investigate the properties of the mock-Chebyshev nodes and propose a subsetting method for constructing mock-Chebyshev grids. Moreover, we provide a precise formula for the cardinality of a satisfactory uniform grid. Some numerical experiments using the points obtained by the method are given to show the effectiveness of the proposed method and numerical results are also provided.

A new algorithm based on compressed Legendre polynomials for solving boundary value problems

<abstract><p>In this paper, we discuss a novel numerical algorithm for solving boundary value problems. We introduce an orthonormal basis generated from compressed Legendre polynomials. This basis can avoid Runge phenomenon caused by high-order polynomial approximation. Based on the new basis, a numerical algorithm of two-point boundary value problems is established. The convergence and stability of the method are proved. The whole analysis is also applicable to higher order equations or equations with more complex boundary conditions. Four numerical examples are tested to illustrate the accuracy and efficiency of the algorithm. The results show that our algorithm have higher accuracy for solving linear and nonlinear problems.</p></abstract>

On the Development of a Synchronized Harmonic Balance Method for Multiple Frequencies and its Application to LPT Flows

The aim of this paper is to describe the development and application of a multi-frequency harmonic balance solver for GPUs, particularly suitable for the simulation of periodic unsteadiness in nonlinear turbomachinery flows comprised of a few dominant frequencies, with an unsteady multistage coupling that bolsters the flow continuity across the rotor/stator interface. The formulation is addressed with the time-domain reinterpretation, where several non-equidistant time instants conveniently selected are solved simultaneously. The set of required frequencies in each row is driven into the governing equations with the help of almost-periodic Fourier transforms for time derivatives and time shifted boundary conditions. The spatial repetitiveness inside each row can be exploited to perform single-passage simulations and the relative circumferential positioning of the rotors or stators and the different blade or vane counts is tackled by means of adding fictitious frequencies referring to non-adjacent rows therefore taking into account clocking and indexing effects. Existing multistage row coupling techniques of harmonic methods rely on the use of non-reflecting boundary conditions, based on linearizations, or time interpolation, which may lead to Runge phenomenon with the resulting numerical instabilities and non-preserving flux exchange. Different sets of time instants might be selected in each row but the interpolation in space and time across their interfaces gives rise to robustness issues due to this phenomenon. The so-called synchronized approach, developed in this work, consist of having the same time instances among the whole ensemble of rows, ensuring that flux transfer at sliding planes is applied more robustly. The combination of a set of shared non-equidistant time instances plus the use of unequal frequencies (real and fictitious) may spoil the Fourier transforms conditioning but this can be dramatically improved with the help of oversampling and instants selection optimization. The resulting multistage coupling naturally addresses typical numerical issues such as flow that might reverse locally across the row interfaces by means of not using boundary conditions but a local flux conservation scheme in the sliding planes. Some examples will be given to illustrate the ability of this new approach to preserve accuracy and robustness while resolving them. A brief analysis of results for a fan stage and a LPT multi-row case is presented to demonstrate the correctness of the method, assessing the impact in the modeling accuracy of the present approach compared with a time-domain conventional analysis. Regarding the computational performance, the speedup compared to a full annulus time-domain unsteady simulation is a factor of order 30 combining the use of single-passage rows and time spectral accuracy.

Euler–Maclaurin expansions without analytic derivatives

The Euler–Maclaurin (EM) formulae relate sums and integrals. Discovered nearly 300 years ago, they have lost none of their importance over the years, and are nowadays routinely taught in scientific computing and numerical analysis courses. The two common versions can be viewed as providing error expansions for the trapezoidal rule and for the midpoint rule, respectively. More importantly, they provide a means for evaluating many infinite sums to high levels of accuracy. However, in all but the simplest cases, calculating very high-order derivatives analytically becomes prohibitively complicated. When approximating such derivatives with finite differences (FD), the choice of step size typically requires a severe trade-off between errors due to truncation and to rounding. We show here that, in the special case of EM expansions, FD approximations can provide excellent accuracy without the step size having to go to zero. While FD approximations of low-order derivatives to high orders of accuracy have many applications for solving ODEs and PDEs, the present context is unusual in that it relies on FD approximations to derivatives of very high orders. The application to infinite sums ensures that one can use centred FD formulae (which are not subject to the Runge phenomenon).

Rational approximation in initial boundary problems with the fronts

Предложен спектральный метод на основе дробно-рациональной аппроксимации. На примере решений уравнения Бюргерса с особенностями в виде фронтов показано, что дробно-рациональное приближение решений имеет существенные преимущества перед полиномиальным. Для эффективной реализации дробно- рациональной аппроксимации в работе использована барицентрическая интерполяционная формула Лагранжа, обеспечивающая быстроту вычислений и численную устойчивость. Для адаптации узлов интерполяции использован метод, основанный на аппроксимации положения особенности аналитического продолжения решения в комплексной плоскость. Предложено обобщение метода на случай нескольких особенностей. Описано построение спектрального метода и проведены расчеты на модельных задачах, в т. ч. с двумя фронтами. A spectral method with adaptive rational approximation is proposed. In traditional spectral polynomial interpolation, the interpolation points are fixed, usually at the roots or extrema of orthogonal polynomials. Free selection of interpolation points is impossible due to the effect described in the Runge example. The key feature of rational interpolation is the free distribution of interpolation nodes without the occurrence of the Runge phenomenon. Nevertheless, in practice it is very important to implement rational approximation effectively. Here rational approximation is implemented using the barycentric Lagrange form. This leads to fast computations and numerical stability comparable with the polynomial interpolation. It is shown that rational interpolation has significant advantages over polynomial on functions that have singularities in the form of fronts. The key idea is that rational interpolation allows adapting interpolation points according to function singularities. An effective method of grid adaptation that accounts for singularity location was used. Method was generalized to the case of several singularities, for example, for solutions with several fronts. For the solutions of the Burgers equation with singularities in the form of fronts, it is shown that rational interpolation has significant advantages over polynomial. The implementation of spectral method is described, and calculations results on model problems, including problems with two fronts, are presented.

Derived Nodes Approach for Improving Accuracy of Machining Stability Prediction

Machining process dynamics can be described by state-space delayed differential equations (DDEs). To numerically predict the process stability, diverse piecewise polynomial interpolation is often utilized to discretize the continuous DDEs into a set of linear discrete equations. The accuracy of discrete approximation of the DDEs generally depends on how to deal with the piecewise polynomials. However, the improvement of the stability prediction accuracy cannot be always guaranteed by higher-order polynomials due to the Runge phenomenon. In this study, the piecewise polynomials with derivative-continuous at joint nodes are taken into consideration. We develop a recursive estimation of derived nodes for interpolation approximation of the state variables, so as to improve the discretization accuracy of the DDEs. Two different temporal discretization methods, i.e., second-order full-discretization and state-space temporal finite methods, are taken as demonstrations to illustrate the effectiveness of applying the proposed approach for accuracy improvement. Numerical simulations prove that the proposed approach brings a great improvement on the accuracy of the stability lobes, as well as the rate of convergence, compared to the previous recorded ones with the same order of interpolation polynomials.

A New Method to Solve Numeric Solution of Nonlinear Dynamic System

It is well known that the cubic spline function has advantages of simple forms, good convergence, approximation, and second-order smoothness. A particular class of cubic spline function is constructed and an effective method to solve the numerical solution of nonlinear dynamic system is proposed based on the cubic spline function. Compared with existing methods, this method not only has high approximation precision, but also avoids the Runge phenomenon. The error analysis of several methods is given via two numeric examples, which turned out that the proposed method is a much more feasible tool applied to the engineering practice.