Optimal control of non‐linear chemical reactors via an initial‐value Hamiltonian problem

The problem of designing strategies for optimal feedback control of non‐linear processes, specially for regulation and set‐point changing, is attacked in this paper. A novel procedure based on the Hamiltonian equations associated to a bilinear approximation of the dynamics and a quadratic cost is presented. The usual boundary‐value situation for the coupled state–costate system is transformed into an initial‐value problem through the solution of a generalized algebraic Riccati equation. This allows to integrate the Hamiltonian equations on‐line, and to construct the feedback law by using the costate solution trajectory. Results are shown applied to a classical non‐linear chemical reactor model, and compared against suboptimal bilinear‐quadratic strategies based on power series expansions. Since state variables calculated from Hamiltonian equations may differ from the values of physical states, the proposed control strategy is suboptimal with respect to the original plant. Copyright © 2005 John Wiley & Sons, Ltd.     

An Interface-Fitted Finite Element Level Set Method with Application to Solidification and Solvation

A new finite element level set method is developed to simulate the interface motion. The normal velocity of the moving interface can depend on both the local geometry, such as the curvature, and the external force such as that due to the flux from both sides of the interface of a material whose concentration is governed by a diffusion equation. The key idea of the method is to use an interface-fitted finite element mesh. Such an approximation of the interface allows an accurate calculation of the solution to the diffusion equation. The interface-fitted mesh is constructed from a base mesh, a uniform finite element mesh, at each time step to explicitly locate the interface and separate regions defined by the interface. Several new level set techniques are developed in the framework of finite element methods. These include a simple finite element method for approximating the curvature, a new method for the extension of normal velocity, and a finite element least-squares method for the reinitialization of level set functions. Application of the method to the classical solidification problem captures the dendrites. The method is also applied to the molecular solvation to determine optimal solute-solvent interfaces of solvation systems.     

Simulating an Elastic Ring with Bend and Twist by an Adaptive Generalized Immersed Boundary Method

Many problems involving the interaction of an elastic structure and a viscous fluid can be solved by the immersed boundary (IB) method. In the IB approach to such problems, the elastic forces generated by the immersed structure are applied to the surrounding fluid, and the motion of the immersed structure is determined by the local motion of the fluid. Recently, the IB method has been extended to treat more general elasticity models that include both positional and rotational degrees of freedom. For such models, force and torque must both be applied to the fluid. The positional degrees of freedom of the immersed structure move according to the local linear velocity of the fluid, whereas the rotational degrees of freedom move according to the local angular velocity. This paper introduces a spatially adaptive, formally second-order accurate version of this generalized immersed boundary method. We use this adaptive scheme to simulate the dynamics of an elastic ring immersed in fluid. To describe the elasticity of the ring, we use an unconstrained version of Kirchhoff rod theory. We demonstrate empirically that our numerical scheme yields essentially second-order convergence rates when applied to such problems. We also study dynamical instabilities of such fluid-structure systems, and we compare numerical results produced by our method to classical analytic results from elastic rod theory.     

An Energy-Preserving Wavelet Collocation Method for General Multi-Symplectic Formulations of Hamiltonian PDEs

In this paper, we develop a novel energy-preserving wavelet collocation method for solving general multi-symplectic formulations of Hamiltonian PDEs. Based on the autocorrelation functions of Daubechies compactly supported scaling functions, the wavelet collocation method is conducted for spatial discretization. The obtained semi-discrete system is shown to be a finite-dimensional Hamiltonian system, which has an energy conservation law. Then, the average vector field method is used for time integration, which leads to an energy-preserving method for multi-symplectic Hamiltonian PDEs. The proposed method is illustrated by the nonlinear Schrödinger equation and the Camassa-Holm equation. Since differentiation matrix obtained by the wavelet collocation method is a cyclic matrix, we can apply Fast Fourier transform to solve equations in numerical calculation. Numerical experiments show the high accuracy, effectiveness and conservation properties of the proposed method.     

A Discrete Flux Scheme for Aerodynamic and Hydrodynamic Flows

The objective of this paper is to seek an alternative to the numerical simulation of the Navier-Stokes equations by a method similar to solving the BGK-type modeled lattice Boltzmann equation. The proposed method is valid for both gas and liquid flows. A discrete flux scheme (DFS) is used to derive the governing equations for two distribution functions; one for mass and another for thermal energy. These equations are derived by considering an infinitesimally small control volume with a velocity lattice representation for the distribution functions. The zero-order moment equation of the mass distribution function is used to recover the continuity equation, while the first-order moment equation recovers the linear momentum equation. The recovered equations are correct to the first order of the Knudsen number (Kn); thus, satisfying the continuum assumption. Similarly, the zero-order moment equation of the thermal energy distribution function is used to recover the thermal energy equation. For aerodynamic flows, it is shown that the finite difference solution of the DFS is equivalent to solving the lattice Boltzmann equation (LBE) with a BGK-type model and a specified equation of state. Thus formulated, the DFS can be used to simulate a variety of aerodynamic and hydrodynamic flows. Examples of classical aeroacoustics, compressible flow with shocks, incompressible isothermal and non-isothermal Couette flows, stratified flow in a cavity, and double diffusive flow inside a rectangle are used to demonstrate the validity and extent of the DFS. Very good to excellent agreement with known analytical and/or numerical solutions is obtained; thus lending evidence to the DFS approach as an alternative to solving the Navier-Stokes equations for fluid flow simulations.     

Numerical Simulation of Free Surface by an Area-Preserving Level Set Method

We apply in this study an area preserving level set method to simulate gas/water interface flow. For the sake of accuracy, the spatial derivative terms in the equations of motion for an incompressible fluid flow are approximated by the fifth-order accurate upwinding combined compact difference (UCCD) scheme. This scheme development employs two coupled equations to calculate the first- and second-order derivative terms in the momentum equations. For accurately predicting the level set value, the interface tracking scheme is also developed to minimize phase error of the first-order derivative term shown in the pure advection equation. For the purpose of retaining the long-term accurate Hamiltonian in the advection equation for the level set function, the time derivative term is discretized by the sixth-order accurate symplectic Runge-Kutta scheme. Also, to keep as a distance function for ensuring the front having a finite thickness for all time, the re-initialization equation is used. For the verification of the optimized UCCD scheme for the pure advection equation, two benchmark problems have been chosen to investigate in this study. The level set method with excellent area conservation property proposed for capturing the interface in incompressible fluid flows is also verified by solving the dam-break, Rayleigh-Taylor instability, two-bubble rising in water, and droplet falling problems.     

Local Discrete Velocity Grids for Multi-Species Rarefied Flow Simulations

This article deals with the derivation of an adaptive numerical method for mono-dimensional kinetic equations for gas mixtures. For classical deterministic kinetic methods, the velocity domain is chosen accordingly to the initial condition. In such methods, this velocity domain is the same for all time, all space points and all species. The idea developed in this article relies on defining velocity domains that depend on space, time and species. This allows the method to locally adapt to the support of the distribution functions. The method consists in computing macroscopic quantities by the use of conservation laws, which enables the definition of such local grids. Then, an interpolation procedure along with a upwind scheme is performed in order to treat the advection term, and an implicit treatment of the BGK operator allows for the derivation of an AP scheme, where the stability condition is independent of the relaxation rate. The method is then applied to a series of test cases and compared to the classical DVM method.     

Multi-Chambered Dialyzers and Their Efficiencies

The performance of a dialyzer at a given test condition is strongly affected by its design and flow patterns as well as other factors such as membrane permeability, membrane area and liquid-side resistances. Mathematical solutions that describe multichambered dialyzers in both countercurrent and cocurrent dialysis modes are given. In limiting cases, as the number of chambers approaches infinity, these solutions yield a simple equation which is essentially the solution to a cross-flow dialyzer in which blood is unmixed and dialysate fluid is mixed. Currently, it is a rather tedious process to obtain a theoretical dialysance value for a cross-flow dialyzer even though many such dialyzers are widely used. In reality, a dialyzer cannot achieve its theoretical performance level because actual flow patterns and flow distributions are not ideal. The effect of non-ideality as a percentage of channeling of dialysate fluid was accounted for in the performance calculations of various types of dialyzers. The ideal single-chambered dialyzer in the countercurrent mode has the highest theoretical performance. Its performance, however, decreases rapidly as the degree of the non-ideality increases, while multi-chambered dialyzers are relatively insensitive to deviation from the ideal condition.     

Direct-Forcing Immersed Boundary Method for Mixed Heat Transfer

A direct-forcing immersed boundary method (DFIB) with both virtual force and heat source is developed here to solve Navier-Stokes and the associated energy transport equations to study some thermal flow problems caused by a moving rigid solid object within. The key point of this novel numerical method is that the solid object, stationary or moving, is first treated as fluid governed by Navier-Stokes equations for velocity and pressure, and by energy transport equation for temperature in every time step. An additional virtual force term is then introduced on the right hand side of momentum equations in the solid object region to make it act exactly as if it were a solid rigid body immersed in the fluid. Likewise, an additional virtual heat source term is applied to the right hand side of energy equation at the solid object region to maintain the solid object at the prescribed temperature all the time. The current method was validated by some benchmark forced and natural convection problems such as a uniform flow past a heated circular cylinder, and a heated circular cylinder inside a square enclosure. We further demonstrated this method by studying a mixed convection problem involving a heated circular cylinder moving inside a square enclosure. Our current method avoids the otherwise requested dynamic grid generation in traditional method and shows great efficiency in the computation of thermal and flow fields caused by fluid-structure interaction.     

The role of symplectic integrators in optimal control

For general optimal control problems, Pontryagin's maximum principle gives necessary optimality conditions, which are in the form of a Hamiltonian differential equation. For its numerical integration, symplectic methods are a natural choice. This article investigates to which extent the excellent performance of symplectic integrators for long‐time integrations in astronomy and molecular dynamics carries over to problems in optimal control. Numerical experiments supported by a backward error analysis show that for problems in low dimension close to a critical value of the Hamiltonian, symplectic integrators have a clear advantage. This is illustrated using the Martinet case in sub‐Riemannian geometry. For problems like the orbital transfer of a spacecraft or the control of a submerged rigid body, such an advantage cannot be observed. The Hamiltonian system is a boundary value problem and the time interval is in general not large enough so that symplectic integrators could benefit from their structure preservation of the flow. Copyright © 2008 John Wiley & Sons, Ltd.     

Contractility of the urinary bladder.

Measurements have been made in vitro of the active mechanical properties of complete pig bladders, electrically stimulated to contract. The results are described with the aid of a model of the bladder wall consisting of a contractile element in series with an elastic element. For the contractile element the active force depends on the velocity of shortening. This relation is well described by a classical Hill equation, provided force is normalized by dividing it by the isometric force at the same bladder volume. The force-extension relation of the series elastic element is non-linear and can be described by an elastic modulus which depends monoexponentially on the extension. In the light of these findings the limitations of existing clinical methods of assessing bladder contractility, and the possibility of developing new methods, are discussed.     

A Stable Arbitrarily High Order Time-Stepping Method for Thermal Phase Change Problems

Thermal phase change problems are widespread in mathematics, nature, and science. They are particularly useful in simulating the phenomena of melting and solidification in materials science. In this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase change model, which is the coupling of a heat transfer equation and a phase field equation. The unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes. A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step, with an efficient scheme of sufficient accuracy to calculate the solution at the first step. It is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time steps. Adaptive time step size strategies can be applied to further benefit from this unconditional stability. Numerical experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.     

On the Wall Shear Stress Gradient in Fluid Dynamics

The gradient of the fluid stresses exerted on curved boundaries, conventionally computed in terms of directional derivatives of a tensor, is here analyzed by using the notion of intrinsic derivative which represents the geometrically appropriate tool for measuring tensor variations projected on curved surfaces. Relevant differences in the two approaches are found by using the classical Stokes analytical solution for the slow motion of a fluid over a fixed sphere and a numerically generated three dimensional dynamical scenario. Implications for theoretical fluid dynamics and for applied sciences are finally discussed.     

An Adjoint State Method for Numerical Approximation of Continuous Traffic Congestion Equilibria

The equilibrium metric for minimizing a continuous congested traffic model is the solution of a variational problem involving geodesic distances. The continuous equilibrium metric and its associated variational problem are closely related to the classical discrete Wardrop's equilibrium. We propose an adjoint state method to numerically approximate continuous traffic congestion equilibria through the continuous formulation. The method formally derives an adjoint state equation to compute the gradient descent direction so as to minimize a nonlinear functional involving the equilibrium metric and the resulting geodesic distances. The geodesic distance needed for the state equation is computed by solving a factored eikonal equation, and the adjoint state equation is solved by a fast sweeping method. Numerical examples demonstrate that the proposed adjoint state method produces desired equilibrium metrics and outperforms the subgradient marching method for computing such equilibrium metrics.     

A Numerical Scheme for the Quantum Fokker-Planck-Landau Equation Efficient in the Fluid Regime

We construct an efficient numerical scheme for the quantum Fokker-Planck-Landau (FPL) equation that works uniformly from kinetic to fluid regimes. Such a scheme inevitably needs an implicit discretization of the nonlinear collision operator, which is difficult to invert. Inspired by work [9] we seek a linear operator to penalize the quantum FPL collision term Q_qFPL in order to remove the stiffness induced by the small Knudsen number. However, there is no suitable simple quantum operator serving the purpose and for this kind of operators one has to solve the complicated quantum Maxwellians (Bose-Einstein or Fermi-Dirac distribution). In this paper, we propose to penalize Q_qFPL by the "classical" linear Fokker-Planck operator. It is based on the observation that the classical Maxwellian, with the temperature replaced by the internal energy, has the same first five moments as the quantum Maxwellian. Numerical results for Bose and Fermi gases are presented to illustrate the efficiency of the scheme in both fluid and kinetic regimes.     

Stabilization of Traffic Flow Based on the Multiple Information of Preceding Cars

To enhance the stability of traffic flow, a new car-following model is proposed by taking into account the support of intelligent transportation system (ITS) information, which includes both the headway and the velocity difference of multiple preceding cars. The new model is based on the Optimal Velocity (OV) model and its extended models. The stability condition of the model is obtained by using the linear stability theory. Through nonlinear analysis, the modified Korteweg-de Vries equation is constructed and solved, and the traffic flow is classified into three types, i.e. stable, metastable, and unstable. The jam phase can thus be described by the kink-antikink soliton solution for the mKdV equation. The numerical simulation results show that compared with previous models considering only one of the ITS information, the proposed model can suppress traffic jams more efficiently when both headway and velocity difference of arbitrary preceding cars are taken into account. The results of numerical simulation coincide with the theoretical ones.     

A Semi-Lagrangian Approach for Dilute Non-Collisional Fluid-Particle Flows

We develop numerical methods for the simulation of laden-flows where particles interact with the carrier fluid through drag forces. Semi-Lagrangian techniques are presented to handle the Vlasov-type equation which governs the evolution of the particles. We discuss several options to treat the coupling with the hydrodynamic system describing the fluid phase, paying attention to strategies based on staggered discretizations of the fluid velocity.     

High-Order Schemes Combining the Modified Equation Approach and Discontinuous Galerkin Approximations for the Wave Equation

We present a new high order method in space and time for solving the wave equation, based on a new interpretation of the "Modified Equation" technique. Indeed, contrary to most of the works, we consider the time discretization before the space discretization. After the time discretization, an additional biharmonic operator appears, which can not be discretized by classical finite elements. We propose a new Discontinuous Galerkin method for the discretization of this operator, and we provide numerical experiments proving that the new method is more accurate than the classical Modified Equation technique with a lower computational burden.     

Novel Symplectic Discrete Singular Convolution Method for Hamiltonian PDEs

This paper explores the discrete singular convolution method for Hamiltonian PDEs. The differential matrices corresponding to two delta type kernels of the discrete singular convolution are presented analytically, which have the properties of high-order accuracy, band-limited structure and thus can be excellent candidates for the spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrödinger equation and the coupled Schrödinger equations for example, we construct two symplectic integrators combining this kind of differential matrices and appropriate symplectic time integrations, which both have been proved to satisfy the square conservation laws. Comprehensive numerical experiments including comparisons with the central finite difference method, the Fourier pseudospectral method, the wavelet collocation method are given to show the advantages of the new type of symplectic integrators.