Framework For Numerical Solution Research Articles

Understanding AP/HTPB composite propellant combustion from new perspectives

Combustion of the AP/HTPB composite propellant at the mesoscale is simulated by a partitioned numerical framework, in which the unsteady solid-phase conduction, gas-solid interface regression, and gaseous combustion are sequentially calculated on two sets of overlapped grids. The gas-solid interface is traced by the level-set method. The mass and species exchange conditions at the gas-solid interface are converted into source terms by the immersed-boundary method. Effectiveness of the entire framework is justified by applying the integrated solver to a Miller pack example. The predicted burning rates show good agreement with experimental data over a wide range of pressure conditions, and the gas-phase temperature field clearly depicts the final diffusion flame. The flame zone closely above the regressing interface can be identified by the fluctuation of the projected velocity components perpendicular to the main stream direction. Statistical results conditional on Lagrangian tracing of fluid parcels released from the AP and binder parts show distinction between the oxidizer-rich and fuel-rich flames in terms of the reaction intensity and the flame structure. Statement of SignificanceFrom the algorithm/simulation perspective: a partitioned numerical solution framework is built upon overlapped structured (for the gas phase) and unstructured (for the solid phase) meshes; the level-set approach is used on the unstructured mesh to identify solid-gas interface with complex geometry. From the analysis perspective: the flame front is newly defined in terms of the tangential velocity components; the fuel/oxidizer evolution pattern is analyzed from conditional statistics based on Lagrangian tracing; new insights in heterogeneous combustion include, for instance, effects of pressure on the tangential velocity fluctuation and flame properties conditional on different ingredients.

Numerical Methods for Fractional Optimal Control and Estimation

Fractional calculus has become a valuable mathematical tool for modeling various physical phenomena exhibiting anomalous dynamics such as memory and hereditary properties. However, the fractional operators lead to difficulties in analysis, optimization, and estimation that limit the application of fractional models. This paper develops numerical methods to solve fractional optimal control and estimation problems with Caputo derivatives of arbitrary order.
 First, fractional Pontryagin's maximum principle is used to formulate first-order necessary conditions for fractional optimal control problems. A fractional collocation method using polynomial basis functions is then proposed to discretize the resulting boundary value problems. This allows transforming an infinite-dimensional optimal control problem into a finite nonlinear programming problem.
 Second, for fractional estimation, a novel ensemble Kalman filter is proposed based on a Monte Carlo approach to propagate the fractional state dynamics. This provides a recursive fractional state estimator analogous to the classical Kalman filter.
 The capabilities of the proposed collocation and ensemble Kalman filter methods are demonstrated through applications including fractional epidemic control, thermomechanical oscillator control, and state estimation of viscoelastic mechanical systems. The results illustrate improved accuracy over prior discretization schemes along with the ability to handle complex system dynamics.
 This work provides a comprehensive framework for numerical solution of fractional optimal control and estimation problems. The methods enable applying fractional calculus to address challenges in robotics, biomedicine, mechanics, and other fields where systems exhibit non-classical dynamics.

Coupled thermo-hydro-mechanical-phase field modeling for fire-induced spalling in concrete

This paper aims at developing an efficient numerical model to numerically reproduce the fire-induced spalling process taking place in concrete and to investigate the underlying mechanism of thermal spalling. To deal with complex crack patterns and with ease of extending to multi-physics problem, phase field method is employed for crack modeling. The primary objective of this manuscript is to develop a coupled thermo-hydro-mechanical-phase field model and the numerical solution framework. The effects of length scale parameter, staggered solution methods, and related factors on mechanical fields, as well as the impact of fracture on mass transfer and heat conduction are investigated. It is concluded that spalling is a result of combined thermal expansion and vapor pressure, among which thermal effect plays a major role and vapor pressure accelerates the occurrence of spalling.

A Unified Probabilistic Discretization Scheme for FBSDEs: Stability, Consistency, and Convergence Analysis

In this work, we propose a general discretization framework for numerical solution of forward backward stochastic differential equations (FBSDEs). The framework covers several existing temporal dis...

Magnetically driven foldable shell type swimmers at Stokes flow

This paper focuses on the interaction of low Reynolds number (Re) flows and thin shell type deformable structures in the context of flexible body locomotion and addresses the coupled field problem through a numerical solution framework. The thin structure is discretized by enhanced three-node finite elements and coupled with boundary element based treatment of Stokes flow in a monolithic manner. The locomotion is triggered and driven by an external magnetic field that generates displacement dependent body couples over the magnetically sensitive parts of the flexible structure. A particular novelty of the paper is the use of internal hinges through which very large rotations and structural deformations can be combined in an efficient way. Using this concept; new, on the fly locomotion direction reversal mechanisms can be generated as demonstrated by the foldable bi-directional swimmer.

A numerical solution framework for simultaneous peeling of thin elastic strips from a rigid substrate

Simultaneous peeling of multiple strips is commonly observed particularly at small-scale detachment processes. Although theoretical treatment of this problem is addressed, numerical solution procedures for geometrically arbitrary multiple-peeling problems are still missing. In this paper, a finite element-based numerical solution procedure for 3-D large displacement multiple-peeling problems is presented. Loading/unloading of peeling strips are expressed in the form of optimality conditions, and the current positions of the peeling fronts are determined locally adapting the multiplicative decomposition and the return mapping algorithm of finite strain plasticity theories. Within an incremental-iterative solution framework, peeling fronts and the current position of other nodes are determined in a staggered way instead of using an active set-based solution algorithm. The effectiveness of the approach is demonstrated by a series of example problems including multiple peeling of an assembly of randomly oriented strips.

Better than pre-commitment mean-variance portfolio allocation strategies: A semi-self-financing Hamilton–Jacobi–Bellman equation approach

We generalize the idea of semi-self-financing strategies, originally discussed in Ehrbar (1990), and later formalized in Cui et al (2012), for the pre-commitment mean-variance (MV) optimal portfolio allocation problem. The proposed semi-self-financing strategies are built upon a numerical solution framework for Hamilton–Jacobi–Bellman equations, and can be readily employed in a very general setting, namely continuous or discrete re-balancing, jump-diffusions with finite activity, and realistic portfolio constraints. We show that if the portfolio wealth exceeds a threshold, an MV optimal strategy is to withdraw cash. These semi-self-financing strategies are generally non-unique. Numerical results confirming the superiority of the efficient frontiers produced by the strategies with positive cash withdrawals are presented. Tests based on estimation of parameters from historical time series show that the semi-self-financing strategy is robust to estimation ambiguities.

Generation of low-frequency higher harmonics in a resonant-tunneling diode

The behavior of the higher harmonics of a resonant-tunneling diode is investigated experimentally and theoretically in the adiabatic limit. Within the framework of numerical solution of the time-dependent Schrodinger equation with open boundary conditions, the currents of the first three harmonics are calculated. It is shown that the behavior of the currents qualitatively agrees with the experimental data and the results of a simple model used by us previously. The possibility of the generation of an electromagnetic field at frequencies multiple to the harmonic number is shown.

Better than Pre-Commitment Mean-Variance Portfolio Allocation Strategies: A Semi-Self- Financing Hamilton-Jacobi-Bellman Equation Approach

We generalize the idea of semi-self-financing strategies, originally discussed in Ehrbar, Journal of Economic Theory (1990), and later formalized in em Cui et al, Mathematical Finance 22 (2012), for the pre-commitment mean-variance (MV) optimal portfolio allocation problem. The proposed semi-self-financing strategies are built upon a numerical solution framework for Hamilton-Jacobi-Bellman equations, and can be readily employed in a very general setting, namely continuous or discrete re-balancing, jump-diffusions with finite activity, and realistic portfolio constraints. We show that if the portfolio wealth exceeds a threshold, an MV optimal strategy is to withdraw cash. These semi-self-financing strategies are generally non-unique. Numerical results confirming the superiority of the efficient frontiers produced by the strategies with positive cash withdrawals are presented.Tests based on estimation of parameters from historical time series show that the semi-self-financing strategy is robust to estimation ambiguities.

Nonlinear propagation of light in structured media: Generalized unidirectional pulse propagation equations

Unidirectional pulse propagation equations [UPPE, Phys. Rev. E 70, 036604 (2004)] have provided a theoretical underpinning for computer-aided investigations into dynamics of high-power ultrashort laser pulses and have been successfully utilized for almost a decade. Unfortunately, they are restricted to applications in bulk media or, with additional approximations, to simple waveguide geometries in which only a few guided modes can approximate the propagating waveform. The purpose of this work is to generalize the directional pulse propagation equations to structures characterized by strong refractive index differences and material interfaces. We also outline a numerical solution framework that draws on the combination of the bulk-media UPPE method with single-frequency beam-propagation techniques.

Numerical Solution Framework of Kinematics for a Tendon-Driven Manipulator Equipped with Cylindrical Elastic Elements

To date, many hardware devices to control joint stiffness have been proposed for tendon-driven manipulators. However, the earlier devices presented some problems such as complex structures, increase of friction, increase of inertia, etc. To overcome these problems, we propose the cylindrical elastic element (CEE) to vary the joint stiffness by changing the internal force among wires. By inserting the CEEs into routes of wires, the joint stiffness of a tendon-driven manipulator can be changed depending on the internal force. However, it is difficult to obtain the kinematics because of the complexity of a CEE's deformation. Since a finite element method usually requires much time to calculate, the establishment of a simple CEE model is very important to solve the kinematics. This paper presents a numerical framework to approximately solve the kinematics of a one-link manipulator equipped with two CEEs. First, we propose some approximate models of a CEE and evaluate them. Using the most useful model, we then demonstrate a numerical solving method of the forward kinematics. Next, expanding this solving method, we also provide the inverse kinematics. The precision of the proposed methods is discussed through comparison of experimental results with numerical results.

Nonlinear Model Predictive Control Using Multiple Shooting Combined with Collocation on Finite Elements

A new approach to nonlinear model predictive control (NMPC) is proposed in this paper. The multiple shooting method is used for discretizing the dynamic system, through which the optimal control problem is transformed to a nonlinear program (NLP). To solve this NLP problem state variables and their gradients at the end of each shooting need to be computed. Here we employ the method of collocation on finite elements to carry out this task. Due to its high numerical accuracy, the computation efficiency for the integration of model equations can be enhanced, in comparison to the existing multiple shooting method where an ODE solver is applied for the integration and the chain-rule for the gradient computation. The numerical solution framework is implemented in C++. Two examples are taken to demonstrate the effectiveness of the proposed NMPC algorithm.

Solution of population balances with breakage and agglomeration by high-order moment-conserving method of classes

A general high-order method of classes framework for numerical solution of population balances is developed and tested. It is based on conservation of an arbitrary number of moments in the numerical discretization of the integrodifferential population balance equation. Optimal construction of the product tables for agglomeration and breakage events are discussed separately. It is shown that if more than two moments are set to be conserved in the numerical scheme, accuracy can be improved by several orders of magnitude compared to the state-of-the-art methods. Several test cases related to pure breakage or agglomeration, and also to simultaneous breakage and agglomeration, are shown. The results show that the present method is extremely accurate for all tested cases already with a limited number of size categories.

Analysis of stochastic Petri nets by the method of supplementary variables

This paper presents a unified numerical solution framework for stochastic Petri nets in which transition firing is immediate, exponentially distributed, or generally distributed. The proposed solution approach is based on the method of supplementary variables and extends the class of nets for which a numerical analysis is possible. Deterministic and Stochastic Petri Nets (DSPN) are taken as the basic modeling formalism which is extended by other non-exponential firing times and concurrently enabled deterministic transition firings. Under the assumption that the non-exponential distributions belong to the class of polynomial distributions, computational formulas for numerically solving the state equations are given. In case of concurrently enabled deterministic transitions the solution approach yields a system of partial differential equations which we propose to solve numerically by discretization.