In this paper we deal with construction and analysis of a multiwavelet spectral element scheme for a generalized Cauchy type problem with Caputo fractional derivative. Numerical schemes for this type of problems, often suffer from the draw-back of spurious oscillations. A common remedy is to render the problem to an equivalent integral equation. For the generalized Cauchy type problem, a corresponding integral equation is of nonlinear Volterra type. In this paper we investigate wellposedness and convergence of a stabilizing multiwavelet scheme for a, one-dimensional case (in [a, b] or [0, 1]), of this problem. Based on multiwavelets, we construct an approximation procedure for the fractional integral operator that yields a linear system of equations with sparse coefficient matrix. In this setting, choosing an appropriate threshold, the number of non-zero coefficients in the system is substantially reduced. A severe obstacle in the convergence analysis is the lack of continuous derivatives in the vicinity of the inflow/ starting boundary point. We overcome this issue through separating a J (mesh)-dependent, small, neighborhood of a (or origin) from the interval, where we only take $$L_2$$L 2 -norm. The estimate in this part relies on Chebyshev polynomials, viz. As reported by Richardson( Chebyshev interpolation for functions with endpoint singularities via exponential and double-exponential transforms, Oxford University, UK, 2012) and decreases, almost, exponentially by raising J. At the remaining part of the domain the solution is sufficiently regular to derive the desired optimal error bound. We construct such a modified scheme and analyze its wellposedness, efficiency and accuracy. The robustness of the proposed scheme is confirmed implementing numerical examples.

This paper presents a nonstandard local approach to Richardson extrapolation, when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in $\mathbb R^N$ , $N \ge 2$ . The main feature of the approach is that it does not rely on a traditional asymptotic error expansion, but rather depends on a more easily proved weaker a priori estimate, derived in [19], called an asymptotic error expansion inequality. In order to use this inequality to verify that the Richardson procedure works at a point, we require a local condition which links the different subspaces used for extrapolation. Roughly speaking, this condition says that the subspaces are similar about a point, i.e., any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point. Examples of finite element subspaces that occur in practice and satisfy this condition are given.

In the automotive industry, corrosion protected galvanized advanced high strength steels with high ductility (AHSS-HD) gain importance due to their good formability and their lightweight potential. Unfortunately, under specific thermomechanical loading conditions such as during resistance spot welding galvanized, AHSS-HD sheets tend to show liquid metal embrittlement (LME). LME is an intergranular decohesion phenomenon leading to a drastic loss of ductility of up to 95%. The occurrence of LME for a given galvanized material mainly depends on thermal and mechanical loading. These influences are investigated for a dual phase steel with an ultimate tensile strength of 1200 MPa, a fracture strain of 14% and high ductility (DP1200HD) by means of systematic isothermal hot tensile testing on a Gleeble^® 3800 thermomechanical simulator. Based on the experimental findings, a machine learning procedure using symbolic regression is applied to calibrate an LME damage model that accounts for the governing quantities of temperature, plastic strain and strain rate. The finite element (FE) implementation of the damage model is validated based on the local damage distribution in the hot tensile tested samples and in an exemplary 2-sheet resistance spot weld. The developed LME damage model predicts the local position and the local intensity of liquid metal induced cracking in both cases very well.

In this work, we present and test an approach based on an inverse model applicable to the control of induction heat treatments. The inverse model is comprised of a simplified analytical forward model trained with experiments to predict and control the temperature of a location in a cylindrical sample starting from any initial temperature. We solve the coupled nonlinear electromagnetic-thermal problem, which contains a temperature dependent parameter $\alpha$ to correct the electromagnetic field on the surface of a cylinder, and as a result effectively the modeled temperature elsewhere in the sample. A calibrated model to the measurement data applied with the process information such as the operating power level, current, frequency, and temperature provides the basic ingredients to construct an inverse model toolbox, which finally enables us to conduct experiments with more specific goals. The input set values of the power supply, i.e., the power levels in the test rig control system, are determined within an iterative framework to reach specific target temperatures in prescribed times. We verify the concept on an induction heating test rig and provide two examples to illustrate the approach. The advantages of the method lie in its simplicity, computationally cost effectiveness and independence of a prior knowledge of the internal structure of power supplies.

Stability and convergence analyses for the domain decomposition finite element/finite difference (FE/FD) method are presented. The analyses are developed for a semi-discrete finite element scheme for time-dependent Maxwell’s equations. The explicit finite element schemes in different settings of the spatial domain are constructed and a domain decomposition algorithm is formulated. Several numerical examples validate convergence rates obtained in the theoretical studies.