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Fokker–Planck equation appears in various areas in natural science, it is used to describe solute transport and Brownian motion of particles. This paper aims to present an efficient and convenient numerical algorithm for space-time fractional differential equations of the Fokker–Planck type.

The main idea of the presented algorithm is to combine polynomials function approximation and fractional differential operator matrices to reduce the studied complex equations to easily solved algebraic equations.

Based on Taylor basis, simple and useful fractional differential operator matrices of alternative Legendre polynomials can be quickly obtained, by which the studied space-time fractional partial differential equations can be transformed into easily solved algebraic equations. Numerical examples and error date are presented to illustrate the accuracy and efficiency of this technique.

Various numerical methods are proposed in complex steps and are computationally expensive. However, the advantage of this paper is its convenient technique, i.e. using the simple fractional differential operator matrices of polynomials, numerical solutions can be quickly obtained in high precision. Presented numerical examples can also indicate that the technique is feasible for this kind of fractional partial differential equations.

This paper aims to present a methodology for volumetric error compensation. This technique is applied to an Objet Eden350V 3D printer and involves a custom measurement strategy.

The kinematic model of the printer is explained, and its error model is simplified to 18 independent error functions. Each error function is defined by a cubic Legendre polynomial. The coefficients of the polynomials are obtained through a Levenberg–Marquardt optimization process. This optimization process compares, in an iterative algorithm, nominal coordinates with actual values of the cloud of points. The points are built in the faces of a gauge artefact as conical sockets defining one unique point for each socket. These points are measured by a coordinate measuring machine self-centring measurement process.

Most of the errors of the 3D printer are systematic. It is possible to obtain an improvement of 70 per cent in terms of global mean error reduction in single points within a volume of 120 × 120 × 40 mm. The forecast of the final error compensation fully matches the actual final error.

This methodology can be used for accuracy improvement in additive manufacturing machines.

Unlike the calculation of geometric errors, the proposed parametric determination through optimization of the error model allows global error reduction, which decreases all sort of systematic errors concurrently. The proposed measurement strategy allows high reliability, high speed and operator independence in the measurement process, which increases efficiency and reduces the cost. The proposed methodology is easily translated to other rapid prototyping machines and allows scalability when replicating artefacts covering any working volume.

The purpose of this paper is to report the implementation of an alternative time integration procedure for the dynamic non-linear analysis of structures.

The time integration algorithm discussed in this work corresponds to a spectral decomposition technique implemented in the time domain. As in the case of the modal decomposition in space, the numerical efficiency of the resulting integration scheme depends on the possibility of uncoupling the equations of motion. This is achieved by solving an eigenvalue problem in the time domain that only depends on the approximation basis being implemented. Complete sets of orthogonal Legendre polynomials are used to define the time approximation basis required by the model.

A classical example with known analytical solution is presented to validate the model, in linear and non-linear analysis. The efficiency of the numerical technique is assessed. Comparisons are made with the classical Newmark method applied to the solution of both linear and non-linear dynamics. The mixed time integration technique presents some interesting features making very attractive its application to the analysis of non-linear dynamic systems. It corresponds in essence to a modal decomposition technique implemented in the time domain. As in the case of the modal decomposition in space, the numerical efficiency of the resulting integration scheme depends on the possibility of uncoupling the equations of motion.

One of the main advantages of this technique is the possibility of considering relatively large time step increments which enhances the computational efficiency of the numerical procedure. Due to its characteristics, this method is well suited to parallel processing, one of the features that have to be conveniently explored in the near future.

Piezoelectric actuators and sensors are an invaluable part of lightweight designs for several reasons. They can either be used in noise cancellation devices as thin‐walled structures are prone to acoustic emissions, or in shape control approaches to suppress unwanted vibrations. Also in Lamb wave based health monitoring systems piezoelectric patches are applied to excite and to receive ultrasonic waves. The purpose of this paper is to develop a higher order finite element with piezoelectric capabilities in order to simulate smart structures efficiently.

In the paper the development of a new fully three‐dimensional piezoelectric hexahedral finite element based on the p‐version of the finite element method (FEM) is presented. Hierarchic Legendre polynomials in combination with an anisotropic ansatz space are utilized to derive an electro‐mechanically coupled element. This results in a reduced numerical effort. The suitability of the proposed element is demonstrated using various static and dynamic test examples.

In the current contribution it is shown that higher order coupled‐field finite elements hold several advantages for smart structure applications. All numerical examples have been found to agree well with previously published results. Furthermore, it is demonstrated that accurate results can be obtained with far fewer degrees of freedom compared to conventional low order finite element approaches. Thus, the proposed finite element can lead to a significant reduction in the overall numerical costs.

To the best of the author's knowledge, no piezoelectric finite element based on the hierarchical‐finite‐element‐method has yet been published in the literature. Thus, the proposed finite element is a step towards a holistic numerical treatment of structural health monitoring (SHM) related problems using p‐version finite elements.

This paper is concerned with static problems, i.e. those which do not change with time. Dynamic problems will be considered in a sequel. The historical development of infinite elements is described. The two main developments, decay function infinite elements and mapped infinite elements, are described in detail. Results obtained using various infinite elements are given, followed by a discussion of possibilities and likely developments.

Adomian's method is completed to obtain the analytic solution of any partial differential equation with boundary conditions defined on the four sides of a rectangle. Adomian's decomposition method is first used to obtain the N‐order approximant to the one direction partial solution that satisfies the boundary conditions on that direction. Then the functions obtained are variationally modified on the four sides to make them compatible with a given experimental error. The product of these transformations is an analytic approximation to the solution which is compatible with both the weak norm imposed on the boundaries and the accuracy imposed in the whole domain.

A general method is proposed to approximate the analytical solution of any time‐dependent partial differential equation with boundary conditions defined on the four sides of a rectangle. To ensure that the approximant satisfies the boundary conditions problem the differential operator is modified with one additional term which takes into account the effect of boundary conditions. Then the new problem can be directly integrated in the same way as an ordinary differential equation. In this work Adomian's decomposition method with analytic extension is used to obtain the first‐order approximant to the solution of a test case. The result is an analytic approximation to the solution which is compatible with both the exact boundary conditions and the accuracy imposed in the whole domain. The solution obtained is compared with the analytic approximation obtained with a Tau‐Legendre spectral method.

The purpose of this paper is to propose a new computational approach for parameter estimation in the Bayesian framework. A posteriori probability density functions are obtained using the polynomial chaos theory for propagating uncertainties through system dynamics. The new method has the advantage of being able to deal with large parametric uncertainties, non‐Gaussian probability densities and nonlinear dynamics.

The maximum likelihood estimates are obtained by minimizing a cost function derived from the Bayesian theorem. Direct stochastic collocation is used as a less computationally expensive alternative to the traditional Galerkin approach to propagate the uncertainties through the system in the polynomial chaos framework.

The new approach is explained and is applied to very simple mechanical systems in order to illustrate how the Bayesian cost function can be affected by the noise level in the measurements, by undersampling, non‐identifiablily of the system, non‐observability and by excitation signals that are not rich enough. When the system is non‐identifiable and an a priori knowledge of the parameter uncertainties is available, regularization techniques can still yield most likely values among the possible combinations of uncertain parameters resulting in the same time responses than the ones observed.

The polynomial chaos method has been shown to be considerably more efficient than Monte Carlo in the simulation of systems with a small number of uncertain parameters. This is believed to be the first time the polynomial chaos theory has been applied to Bayesian estimation.

The purpose of this paper is to present an efficient method for the numerical treatment of robust optimization problems with absolute reliability constraints.

Optimization with anti-optimization based on response surface techniques; polynomial chaos for approximation of the stochastic objective function.

The number of function calls is comparable to that of the corresponding deterministic problem. Thus, the method is well suited for complex technical systems. The performance of the method is demonstrated on an optimal design problem for turbochargers.

The highlights of this paper are: algorithms for robust and deterministic problems show comparable complexity; no derivatives required; good convergence properties because of special set up of optimization problem; application in complex industrial examples.