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This work presents a meshfree particle scheme designed for arbitrary deformations that possess the accuracy and properties of the Finite-Element-Method. The accuracy is maintained even with arbitrary particle distributions. Mesh-based methods mostly fail if requirements on the location of evaluation points are not satisfied. Hence, with this new scheme not only the range of loadings can be increased but also the pre-processing step can be facilitated compared to the FEM. The key to this new meshfree method lies in the fulfillment of essential requirements for spatial discretization schemes. The new approach is based on the correspondence theory of Peridynamics. Some modifications of this framework allows for a consistent and stable formulation. By applying the peridynamic differentiation concept, it is also shown that the equations of the correspondence theory can be derived from the weak form. Likewise, it is demonstrated that special moving least square shape functions possess the Kronecker- δ property. Thus, Dirichlet boundary conditions can be directly applied. The positive performance of this new meshfree method, especially in comparison to the Finite-Element-Method, is shown in the calculation of several test cases. In order to guarantee a fair comparison enhanced finite element formulations are also used. The test cases include the patch test, an eigenmode analysis as well as the investigation of loadings in the context of large deformations.

In this paper, a computational method is proposed for solving a class of fractional optimal control problems subject to canonical constraints of equality and inequality. Fractional derivatives are described in the Atangana–Baleanu-Caputo sense, and their fractional orders can be different. To solve this problem, we present a discretization scheme based on the trapezoidal rule and a novel numerical integration technique. Then, the gradient formulas of the cost and constraint functions with respect to the decision variables are derived. Furthermore, a gradient-based optimization algorithm for solving the discretized optimal control problem is developed. Finally, the applicability and effectiveness of the proposed algorithm are verified through three non-trivial example problems.

The statistical problem of parameter estimation in partially observed hypoelliptic diffusion processes is naturally occurring in many applications. However, because of the noise structure, where the noise components of the different co-ordinates of the multi-dimensional process operate on different timescales, standard inference tools are ill conditioned. We propose to use a higher order scheme to approximate the likelihood, such that the different timescales are appropriately accounted for. We show consistency and asymptotic normality with non-typical convergence rates. When only partial observations are available, we embed the approximation in a filtering algorithm for the unobserved co-ordinates and use this as a building block in a stochastic approximation expectation–maximization algorithm. We illustrate on simulated data from three models: the harmonic oscillator, the FitzHugh–Nagumo model used to model membrane potential evolution in neuroscience and the synaptic inhibition and excitation model used for determination of neuronal synaptic input.

This paper considers an optimal control problem governed by nonlinear fractional-order systems with multiple time-varying delays and subject to canonical constraints, where the fractional-order derivatives are expressed in the Caputo sense. To solve the problem by discretization scheme, an explicit numerical integration technique is proposed for solving the fractional-order system, and the trapezoidal rule is introduced to approximate the cost functional. Then, the gradients of the resulting cost and constraint functions are derived. On the basis of this result, we develop a gradient-based optimization algorithm to numerically solve the discretized problem. Finally, numerical results of several non-trivial examples are provided to illustrate the applicability and effectiveness of the proposed algorithm.

Computational Fluid Dynamics (CFD) has become an indispensable tool that can potentially predict many phenomena of practical interest in the tundish. Model verification and validation (V&V) are essential parts of a CFD model development process if the models are to be used with sufficient confidence in real industrial tundish applications. The crucial aspects of CFD simulations in the tundish are addressed in this study, such as the selection of the turbulence models, meshing, boundary conditions, and selection of discretization schemes. A series of CFD benchmarking exercises are presented serving as selected examples of appropriate modelling strategies. A tundish database, initiated by German Steel Institute VDEH working group “Fluid Mechanics and Fluid Simulation”, was revisited with the aim of establishing a comprehensive set of best practice guidelines (BPG) in CFD simulations for tundish applications. These CFD benchmark exercises yield important results for the sensible application of CFD models and contribute to further improving the reliability of CFD applications in metallurgical reactors.

The effect of rotation-curvature correction and inviscid spatial discretization scheme on the aerodynamic performance and flow characteristics of Darrieus H-type vertical axis wind turbine (VAWT) are investigated based on an in-house solver. This solver is developed on an in-house platform HRAPIF based on the finite volume method (FVM) with the elemental velocity vector transformation (EVVT) approach. The present solver adopts the density-based method with a low Mach preconditioning technique. The turbulence models are the Spalart-Allmaras (SA) model and the k - ω shear stress transport (SST) model. The inviscid spatial discretization schemes are the third-order monotone upstream-centered schemes for conservation laws (MUSCL) scheme and the fifth-order modified weighted essentially non-oscillatory (WENO-Z) scheme. The power coefficient, instantaneous torque of blades, blade wake, and turbine wake are compared and analyzed at different tip speed ratios. The extensive analysis reveals that the density-based method can be applied in VAWT numerical simulation; the SST models perform better than the SA models in power coefficient prediction; the rotation-curvature correction is not necessary and the third-order MUSCL is enough for power coefficient prediction, the high-order WENO-Z scheme can capture more flow field details, the rotation-curvature correction and high-order WENO-Z scheme reduce the length of the velocity deficit region in the turbine wake.

We propose two stochastic models for the interaction between the myosin head and the actin filament, the physio-chemical mechanism triggering muscle contraction and that is not yet completely understood. We make use of the fractional calculus approach with the purpose of constructing non-Markov processes for models with $ memory. $ A time-changed process and a fractionally integrated process are proposed for the two models. Each of these includes memory effects in a different way. We describe such features from a theoretical point of view and with simulations of sample paths. Mean functions and covariances are provided, considering constant and time-dependent tilting forces by which effects of external loads are included. The investigation of the dwell time of such phenomenon is carried out by means of density estimations of the first exit time (FET) of the processes from a strip; this mimics the times of the Steps of the myosin head during the sliding movement outside a potential well due to the interaction with actin. For the case of time-changed diffusion process, we specify an equation for the probability density function of the FET from a strip. The schemes of two simulation algorithms are provided and performed. Some numerical and simulation results are given and discussed.

Alzheimer's disease is a degenerative disorder characterized by the loss of synapses and neurons from the brain, as well as the accumulation of amyloid-based neuritic plaques. While it remains a matter of contention whether β-amyloid causes the neurodegeneration, β-amyloid aggregation is associated with the disease progression. Therefore, gaining a clearer understanding of this aggregation may help to better understand the disease. We develop a continuous-time model for β-amyloid aggregation using concepts from chemical kinetics and population dynamics. We show the model conserves mass and establish conditions for the existence and stability of equilibria. We also develop two discrete-time approximations to the model that are dynamically consistent. We show numerically that the continuous-time model produces sigmoidal growth, while the discrete-time approximations may exhibit oscillatory dynamics. Finally, sensitivity analysis reveals that aggregate concentration is most sensitive to parameters involved in monomer production and nucleation, suggesting the need for good estimates of such parameters.

This study presents a Complete Discretization Scheme (CDS) for milling stability prediction. When compared with the Semi-Discretization Method (SDM) and Full-Discretization Method (FDM), the highlight of CDS is that it discretizes all parts of Delay Differential Equation (DDE), including delay term, time domain term, parameter matrices, and most of all the differential term, by using the numerical iteration method, such as Euler’s method, to replace the direct integration scheme used in SDM and FDM, which greatly simplifies the complexity of the discretization iteration formula. The present study mainly provides a numerical framework than a method that can be theoretically used by different numerical methods for solving Ordinary Differential Equation (ODE), such as Euler’s method, Runge–Kutta method, Adam’s multistep methods, etc., in this framework for derivation of iteration formula with corresponding construction of coefficient matrix of iteration formula. This study presented CDS with Euler’s method (CDSEM) for solving the one degree-of-freedom problem (one DOF) and two DOF motion equations, which are usually used as benchmark problems. When compared with SDM and FDM, the benchmark results of one DOF and two DOF milling stability prediction show that CDSEM can obtain acceptable precision in most ranges. The computational efficiency of SDM and FDM was also determined, and the results show that CDS with Euler’s method is faster than FDM. Furthermore, large approximation parameters (small time interval) were selected by SDM and CDSEM, and the results show that CDS has high effectiveness, accuracy, and reliability.

The full discrete approximation of solutions of nonhomogeneous fractional equations is considered in this paper. The methods of iteration, finite differences and projection are applied to obtain desired formulas of explicit- and implicit-difference schemes for discretization schemes. The stability of two difference schemes is also discussed using the Trotter–Kato theorem.