An inverse problem formulation of the immersed-boundary method

We formulate the immersed-boundary method (IBM) as an inverse problem. A control variable is introduced on the boundary of a larger domain that encompasses the target domain. The optimal control is the one that minimizes the mismatch between the state and the desired boundary value along the immersed target-domain boundary. We begin by investigating a naïve problem formulation that we show is ill-posed: in the case of the Laplace equation, we prove that the solution is unique, but it fails to depend continuously on the data; for the linear advection equation, even solution uniqueness fails to hold. These issues are addressed by two complimentary strategies. The first strategy is to ensure that the enclosing domain tends to the true domain, as the mesh is refined. The second strategy is to include a specialized parameter-free regularization that is based on penalizing the difference between the control and the state on the boundary. The proposed inverse IBM is applied to the diffusion, advection, and advection-diffusion equations using a high-order discontinuous Galerkin discretization. The numerical experiments demonstrate that the regularized scheme achieves optimal rates of convergence and that the reduced Hessian of the optimization problem has a bounded condition number, as the mesh is refined.     

Parametric variational solution of linear-quadratic optimal control problems with control inequality constraints

A parametric variational principle and the corresponding numerical algo- rithm are proposed to solve a linear-quadratic （LQ） optimal control problem with control inequality constraints. Based on the parametric variational principle, this control prob- lem is transformed into a set of Hamiltonian canonical equations coupled with the linear complementarity equations, which are solved by a linear complementarity solver in the discrete-time domain. The costate variable information is also evaluated by the proposed method. The parametric variational algorithm proposed in this paper is suitable for both time-invariant and time-varying systems. Two numerical examples are used to test the validity of the proposed method. The proposed algorithm is used to astrodynamics to solve a practical optimal control problem for rendezvousing spacecrafts with a finite low thrust. The numerical simulations show that the parametric variational algorithm is ef- fective for LQ optimal control problems with control inequality constraints.     

On the efficiency of a linear controller under additional geometric constraints

One heuristic approach to taking into account geometric constraints on the controls in stabilization problems is to use controls obtained by truncation (at the constraint values) of a control signal linear in the phase variables. With the introduction of the truncated control, the originally linear system becomes substantially nonlinear, which complicates the analysis. In numerous papers, the phase plane method was used to analyze the control defined as the sign of a control signal linear in the phase variables. In [1, 2], the asymptotic stability of linear dynamical systems with nonlinear controls of special type different from that considered below was studied. The problem of stabilization of a mechanical system by a geometrically constrained control was considered in [3]. The asymptotic stability of an arbitrary linear system with a truncated control was studied in [4], where some estimates for the attraction domain of the trivial solution of the system were obtained and necessary and sufficient conditions under which this domain can be made arbitrarily large were given. In the present paper, we solve the problem of ensuring the asymptotic stability of amechanical system with arbitrarily many degrees of freedom and with componentwise geometric constraints on the control.     

结构边界约束修改后的重分析方法

在结构设计优化中经常将结构边界约束作为设计优化对象,结构边界约束的修改通常导致系统的求解规模发生改变,使得快速准确分析修改后结构的响应成为一个挑战。本文发展了逐次矩阵逆(SMI)方法,提出了一种适合各种结构边界约束(包括初始结构中的约束)修改的快速重分析算法。该方法利用边界约束修改后对应刚度矩阵的对称性,有效缩减了计算量。数值算例表明,本文方法能够快速给出精确的重分析结果。     

Feedback minimization of first-passage failure of quasi non-integrable Hamiltonian systems

The non-linear stochastic optimal control of quasi non-integrable Hamiltonian systems for minimizing their first-passage failure is investigated. A controlled quasi non-integrable Hamiltonian system is reduced to an one-dimensional controlled diffusion process of averaged Hamiltonian by using the stochastic averaging method for quasi non-integrable Hamiltonian systems. The dynamical programming equations and their associated boundary and final time conditions for the problems of maximization of reliability and of maximization of mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The dynamical programming equations for maximum reliability problem and for maximum mean first-passage time problem are finalized and their relationships to the backward Kolmogorov equation for the reliability function and the Pontryagin equation for mean first-passage time, respectively, are pointed out. The boundary condition at zero Hamiltonian is discussed. Two examples are worked out to illustrate the application and effectiveness of the proposed procedure.     

Second-Order Fredholm Equations for the First Boundary-Value Problem in the Two-Dimensional Anisotropic Theory of Elasticity

A complete potential theory is constructed for the first boundary-value problem in the two-dimensional anisotropic theory of elasticity (the force vector is specified on the boundary) in a bounded domain on a plane with a Lyapunov boundary. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 85–94, March–April, 2006.     

Principal Compliance and Robust Optimal Design

The paper addresses a problem of robust optimal design of elastic structures when the loading is unknown and only an integral constraint for the loading is given. We propose to minimize the principal compliance of the domain equal to the maximum of the stored energy over all admissible loadings. The principal compliance is the maximal compliance under the extreme, worst possible loading. The robust optimal design is formulated as a min-max problem for the energy stored in the structure. The maximum of the energy is chosen over the constrained class of loadings, while the minimum is taken over the design parameters. It is shown that the problem for the extreme loading can be reduced to an elasticity problem with mixed nonlinear boundary conditions; the last problem may have multiple solutions. The optimization with respect to the designed structure takes into account the possible multiplicity of extreme loadings and divides resources (reinforced material) to equally resist all of them. Continuous change of the loading constraint causes bifurcation of the solution of the optimization problem. It is shown that an invariance of the constraints under a symmetry transformation leads to a symmetry of the optimal design. Examples of optimal design are investigated; symmetries and bifurcations of the solutions are revealed.     

On the Optimal Shape of a Compressed Rotating Rod

By using Pontryagin's maximum principle we determine the shape of the lightest compressed rotating rod, stable against buckling. It is shown that the cross-sectional area function is determined from the solution of a nonlinear boundary value problem. A variational principle for this boundary value problem is formulated and a first integral is constructed. The optimal shape of a rod is determined by numerical integration.     

Solution of the optimal turn problem for a spherically symmetric rigid body with arbitrary boundary conditions in the class of generalized conical motions

We consider the problem of time- and energy consumption-optimal turn of a rigid body with spherical mass distribution under arbitrary boundary conditions on the angular position and angular velocity of the rigid body. The optimal turn problem is modified in the class of generalized conical motions, which allows one to obtain closed-form solutions for equations of motion with arbitrary constants. Thus, solving the optimal control boundary value problem is reduced to solving a system of nonlinear algebraic equations for the constants. Numerical examples are considered to illustrate the proximity between the solutions of the traditional and modified problems of optimal turn of a rigid body.     

Construction of optimal laws of variation of the angular momentum vector of a rigid body

We consider the problem of constructing optimal preset laws of variation of the angular momentum vector of a rigid body taking the body from an arbitrary initial angular position to the required terminal angular position in a given time. We minimize an integral quadratic performance functional whose integrand is a weighted sum of squared projections of the angular momentum vector of the rigid body. We use the Pontryagin maximum principle to derive necessary optimality conditions. In the case of a spherically symmetric rigid body, the problem has a well-known analytic solution. In the case where the body has a dynamic symmetry axis, the obtained boundary value optimization problem is reduced to a system of two nonlinear algebraic equations. For a rigid body with an arbitrarymass distribution, optimal control laws are obtained in the form of elliptic functions. We discuss the laws of controlled motion and applications of the constructed preset laws in systems of attitude control by external control torques or rotating flywheels.     

A three-dimensional optimal control problem in determining the boundary control heat fluxes

A three-dimensional optimal control algorithm in determining the strength of the unknown optimal boundary control heat fluxes utilizing the Conjugate Gradient Method (CGM) and a general purpose commercial code CFX4.2 is applied successfully in the present study based on the desired domain temperature distributions at the final time of heating. Results obtained by using the conjugate gradient method to solve this three-dimensional optimal control problems are justified based on the numerical experiments. Two different computational domains and two different desired temperature distributions are given and the corresponding optimal control heat fluxes are to be determined. Results show that the optimal control heat fluxes can always be obtained with any arbitrary initial guesses of the boundary fluxes.     

Analytical solution of the optimal attitude maneuver problem with a combined objective functional for a rigid body in the class of conical motions

The optimal attitude maneuver control problem without control constraints is studied in the quaternion statement for a rigid body with a spherical mass distribution. The performance criterion is given by a functional combining the time and energy used for the attitude maneuver. A new analytical solution in the class of conical motions is obtained for this problem on the basis of the Pontryagin maximum principle.     

The Optimal Shape of Riblets in the Viscous Sublayer

Our aim is to find the optimal shape of periodically distributed microstructures on surfaces of swimming bodies in order to reduce their drag. The model describes the flow in the viscous sublayer of the boundary layer of a turbulent flow. The microscopic optimization problem is reduced applying homogenization. In the reduced so-called macroscopic optimization problem we minimize the Navier constant subject to the boundary layer equations which are solved in a very small part of the original domain. Under the assumptions that the microstructures can be represented as smooth functions the sensitivity can be determined analytically. The optimization problem is then solved by a sensitivity based method (steepest descent with optimal step size) and the state equations are solved in each iteration with an external software. Our reduced model is validated by comparing the results from the homogenized model with those obtained by simulating the whole rough channel. An improved shape is found and a drag reduction up to 10% can be shown.     

Formation of the impermeable layer in the process of methane hydrate dissociation in porous media

The problem of decomposition of methane hydrate coexisting with water in a highpermeability reservoir is considered. The asymptotic solution is obtained for the decomposition regime in the negative temperature domain. Energy estimates presented show that an impermeable layer saturated with a hydrate-icemixture can be formed in reservoirs with initial positive temperature. The mathematical model of the process of hydrate decomposition is formulated under the assumption on the presence of such a layer in a high-permeability reservoir. In this case the problem is reduced to a purely thermal problem with two unknown moving boundaries. The water-ice phase transition takes place on the leading boundary, while hydrate dissociates at negative temperatures on the slower boundary. The conditions of existence of the layer saturated with a hydrate-ice mixture which is implemented in reservoirs with the high hydrate content are investigated.     

A comparison between a collocation and weak implementation of the rigid‐body motion constraint on a particle surface

In this work, we implemented and compared two different methods to impose the rigid‐body motion constraint on a solid particle moving inside a fluid. We consider a fictitious domain method to easily manage the particle motion. As the solid as well as the fluid inertia are neglected, the particle can be discretized through its boundary only. The rigid‐body motion is imposed via Lagrange multipliers on the boundary. In the first method, such constraints are imposed in discrete points on the boundary (collocation), whereas in the second the constraint is imposed in a weak way on elements dividing the particle surface. Two test problems, that is, a spherical and an ellipsoidal particle in a sheared Newtonian fluid, are chosen to compare the methods. In both cases, the analysis is carried out in 2D as well as in 3D. The results show that for the collocation method an optimal number of collocation points exist leading to the smallest error. However, small variations in the optimal value can generate large deviations. In the weak implementation, the error is only mildly affected by the number of elements used to discretize the particle boundary and by the Lagrange multiplier's interpolation space. A further analysis is carried out to study the effect of an approximated integration of weak constraints. A comparison between the two methods showed that the same accuracy can be achieved by using less constraints if the weak discretization is used. Finally, the rigid‐body motion imposed via weak constraints leads to better conditioned linear systems. Copyright © 2009 John Wiley & Sons, Ltd.     

Construction of Elementary Solutions to a Plane Elastic Problem for a Rectangular Domain

Necessary existence conditions are established for the solution to a plane elastic problem for a rectangular domain given with external boundary and bulk forces. Elementary solutions satisfying Saint-Venant's and superposition principles are constructed.     

Distributed optimal control of a nonstandard system of phase field equations

We investigate a distributed optimal control problem for a phase field model of Cahn–Hilliard type. The model describes two-species phase segregation on an atomic lattice under the presence of diffusion; it has been introduced recently in Colli et?al. (SIAM J Appl Math), on the basis of the theory developed in Podio-Guidugli (Ric. Mat. 55:105–118, 2006), and consists of a system of two highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.     

Non-linear free surface flows and an application of the Orlanski boundary condition

The focus of this paper is the analysis of spatially two-dimensional non-linear free surface problems. The critical aspects of the problem concern the treatment of the non-linear free surface, the body boundary condition for large motions and the imposition of suitable radiation conditions. To address such complexities, time domain simulation was chosen as the method of analysis. With the use of a finite domain for simulation, a major concern is with the radiation condition to be applied at the open or truncation boundary. For the two-dimensional problem at hand, no theoretical radiation conditions are known to exist. An extension of the Orlanski open boundary condition, based on phase velocity determination at the free surface, is proposed. Three categories of problems were analysed using numerical simulation-namely, freely moving steep waves, waves over a submerged body and forced body motion. Simulation results have been compared with linear theory and experiments.     

A qualitative analysis of the brachistochrone problem with dry friction and maximizing the horizontal range

The problem of maximizing the horizontal coordinate of a point moving in a vertical plane under the action of gravity and dry friction and the corresponding brachistochrone problem are considered. The optimal control problem is reduced to a boundary value problem for a system of two nonlinear differential equations. A qualitative analysis of the trajectories of this system is carried out, their typical features are found and illustrated by numerical solving of the boundary value problem. It is shown that the normal component of the support reaction should be positive when moving along the optimal curve. The optimality of the found extremals is discussed.     

Asymptotic models for the discrete optimal control of the deformation of an elastic membrane

This paper considers the singularly perturbed static problem of the optimal control of the deformation of an elastic membrane by means of external loads (control without constraints) applied to several small areas distant from each other. The objective functional is equal to the sum of the square of the root-mean-square approximation error and the square of the norm of the external load. Asymptotic models are constructed using the method of matched asymptotic expansions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 131–144, September–October, 2006.