Mapped Chebyshev Pseudospectral Method Research Articles

Reentry trajectory planning for hypersonic vehicles via an improved sequential convex programming method

The reentry trajectory planning problem of hypersonic vehicles is generally a continuous and nonconvex optimization problem. The solution efficiency and convergence properties of the planning algorithm become an essential point when considering the practical application scenarios. In this paper, an augmented-Lagrange-based improved sequential convex programming algorithm is proposed to solve it and achieve superior solutions with higher accuracy and efficiency. Firstly, the mapped Chebyshev pseudo-spectral method is utilized to convert the continuous optimal control problem into a highly approximately discretized one. In this framework, the discretization technique is employed to discretize the states and control variables at the time interval, which leads to the improvement of ill-conditioned differential matrix and better approximation properties of the variables. Moreover, by introducing slack variables, automatically updated penalty parameters and Lagrange multipliers during the convexification processing, the original nonconvex problem is transformed into an augmented-Lagrange-function-formed relaxed convex one. Then, the arrived convex subproblem can be solved iteratively by the matured interior-point method until the predetermined precision is satisfied. Based on this transformation, it can effectively alleviate the numerical difficulty and improve the convergence property of the proposed improved sequential convex programming algorithm. Finally, Rigorous theoretical analysis is conducted to prove that the solution sequence generated by the proposed algorithm could converge to an approximate global solution of the original problem. Numerical simulations are also presented, and it is demonstrated that the solution time of the proposed algorithm is more than 95% less than that of the traditional nonlinear programming methods without losing the merits of high precision.

Mapped Chebyshev pseudospectral methods for optimal trajectory planning of differentially flat hypersonic vehicle systems

This paper presents a novel trajectory planning algorithm for hypersonic glide vehicles based on differentially flatness theory and mapped Chebyshev pseudospectral method (MCPM). Firstly, the flatness property of a hypersonic glide vehicle dynamics system is extended explicitly by introducing a pseudo control input along the opposite direction of aerodynamics drag. In this way, all the states and controls of the system can be represented by flat outputs and their derivatives, and the trajectory planning problem is formulated into the flat output space with differential constraints entirely eliminated. Secondly, the optimization problem in the flat output space is converted into a nonlinear programming one via the mapped Chebyshev pseudospectral method. The MCPM employs the conformal mapping principle to discretize the flat outputs at the time interval, in terms of which, the closely clustered Chebyshev points near the boundary are stretched and the ill-conditions of the differential matrix is improved. The Barycentric Lagrange interpolation algorithm is then utilized to interpolate the flat output functions, which is seized of better approximation properties than polynomial interpolation. Finally, the transformed nonlinear programming problem are solved by sequence quadratic program technique. The states and controls are then generated by the expressions of the flatness outputs. At the end of this paper, numerical simulations are performed to evaluate the proposed approaches, and comparisons are made with traditional methods. Simulation results demonstrate that the proposed differentially flatness based mapped Chebyshev pseudospectral method possessed better computing efficiency than traditional ones without losing the merits of high precision.

Flatness-based trajectory planning for electromagnetic spacecraft proximity operations in elliptical orbits

The inter-craft electromagnetic force, generated by magnetic coils equipped on all neighboring spacecraft, enables a novel propulsion mode without propellant consumption. The paper investigates a trajectory planning algorithm based on differential flatness for spacecraft proximity operations in elliptical orbits using the inter-craft electromagnetic forces. Firstly, the time-energy optimal trajectory planning problem is formulated into the desired electromagnetic control acceleration optimization and magnetic dipoles assignment. Secondly, utilizing the system's differential flatness property, the original trajectory planning problem could be mapped into the flat output space, eliminating the differential equations and reducing the dimensionality of the planning space. Then the resulted flat output planning problem is further parameterized with a mapped Chebyshev pseudospectral method, which is improved by the conformal map and barycentric rational interpolation techniques in order to decrease the side effect of the differentiation matrix's ill-conditioning. Finally, numerical simulations for a three-spacecraft electromagnetic formation reconfiguration mission show that the proposed algorithm is feasible and efficient, compared with the Radau pseudospectral method.

Prediction of Dynamic Stability Using Mapped Chebyshev Pseudospectral Method

A mapped Chebyshev pseudospectral method is extended to solve three-dimensional unsteady flow problems. As the classical Chebyshev spectral approach can lead to numerical instabilities due to ill conditioning of the spectral matrix, the Chebyshev points are evenly redistributed over the domain by an inverse sine mapping function. The mapped Chebyshev pseudospectral method can be used as an alternative time-spectral approach that uses a Chebyshev collocation operator to approximate the time derivative terms in the unsteady flow governing equations, and the method can make general applications to both nonperiodic and periodic problems. In this study, the mapped Chebyshev pseudospectral method is employed to solve three-dimensional periodic problem to verify the spectral accuracy and computational efficiency with those of the Fourier pseudospectral method and the time-accurate method. The results show a good agreement with both of the Fourier pseudospectral method and the time-accurate method. The flow solutions also demonstrate a good agreement with the experimental data. Similar to the Fourier pseudospectral method, the mapped Chebyshev pseudospectral method approximates the unsteady flow solutions with a precise accuracy at a considerably effective computational cost compared to the conventional time-accurate method.

Mapped Chebyshev Pseudo-Spectral Method for Dynamic Aero-Elastic Problem of Limit Cycle Oscillation

A mapped Chebyshev pseudo-spectral method is developed as one of the Fourier-spectral approaches and solves nonlinear PDE systems for unsteady flows and dynamic aero-elastic problem in a given time interval, where the flows or elastic motions can be periodic, nonperiodic, or periodic with an unknown frequency. The method uses the Chebyshev polynomials of the first kind for the basis function and redistributes the standard Chebyshev–Gauss–Lobatto collocation points more evenly by a conformal mapping function for improved numerical stability. Contributions of the method are several. It can be an order of magnitude more efficient than the conventional finite difference-based, time-accurate computation, depending on the complexity of solutions and the number of collocation points. The method reformulates the dynamic aero-elastic problem in spectral form for coupled analysis of aerodynamics and structures, which can be effective for design optimization of unsteady and dynamic problems. A limit cycle oscillation (LCO) is chosen for the validation and a new method to determine the LCO frequency is introduced based on the minimization of a second derivative of the aero-elastic formulation. Two examples of the limit cycle oscillation are tested: nonlinear, one degree-of-freedom mass–spring–damper system and two degrees-of-freedom oscillating airfoil under pitch and plunge motions. Results show good agreements with those of the conventional time-accurate simulations and wind tunnel experiments.

Mapped Chebyshev pseudo-spectral method for simulating the shear wave propagation in the plane of symmetry of a transversely isotropic viscoelastic medium.

Shear wave elastography is a versatile technique that is being applied to many organs. However, in tissues that exhibit anisotropic material properties, special care must be taken to estimate shear wave propagation accurately and efficiently. A two-dimensional simulation method is implemented to simulate the shear wave propagation in the plane of symmetry in transversely isotropic viscoelastic media. The method uses a mapped Chebyshev pseudo-spectral method to calculate the spatial derivatives and an Adams-Bashforth-Moulton integrator with variable step sizes for time marching. The boundaries of the two-dimensional domain are surrounded by perfectly matched layers to approximate an infinite domain and minimize reflection errors. In an earlier work, we proposed a solution for estimating the apparent shear wave elasticity and viscosity of the spatial group velocity as a function of rotation angle through a low-frequency approximation by a Taylor expansion. With the solver implemented in MATLAB, the simulated results in this paper match well with the theory. Compared to the finite element method simulations we used before, the pseudo-spectral solver consumes less memory and is faster and achieves better accuracy.

Mapped Chebyshev Pseudospectral Method for Unsteady Flow Analysis

A mapped Chebyshev pseudospectral method is developed as an accurate and yet efficient approach to solve unsteady flows. Preserving the conservation laws, the method discretizes a spatial-derivative term implicitly, whereas a time-derivative term is treated explicitly using the mapped Chebyshev collocation operator. Because standard Chebyshev points make the corresponding spectral derivative matrix ill-conditioned due to uneven distribution of the Chebyshev points clustered heavily toward the ends of the interval, an inverse sine mapping function is applied to the Chebyshev collocation operator so that the point distribution becomes more uniform in a time domain and mitigates numerical instabilities correspondingly. Computations of unsteady flows, including the one-dimensional Burgers’ equation and two-dimensional airfoil flows under oscillation and plunging motions, are carried out. Numerical results of the present study are compared with those of the conventional time-marching solution method and the harmonic balance method for each application and demonstrate high accuracy at much reduced computational cost. The method shows great potential to solve general unsteady flows involving both periodic and nonperiodic motions.

Optimal guidance for hypersonic reentry using inversion and receding horizon control

This study investigates the optimal guidance problem for reentry vehicles with focus on novel methods for nominal trajectory generation and tracking. To reduce the dimensionality of reentry trajectory optimisation problem, the inversion-based approach is utilised to partly eliminate the differential equations by reformulating the states and controls with some desired outputs. These outputs are parameterised with a mapped Chebyshev pseudospectral method, which is improved by conformal map and barycentric rational interpolation techniques in order to enhance the numerical accuracy for high-order derivatives of desired outputs. On the basis of the obtained nominal trajectory, a robust closed-loop guidance scheme is proposed using receding horizon control, whereas the associated two-point boundary value problem in each guidance cycle is readily solved by differential transformation method. Numerical simulations show that the proposed guidance scheme is feasible and effective for atmospheric reentry.

Simulation of ultrasound radiation force induced shear wave propagation in viscoelastic media using a mapped Chebyshev pseudospectral method

We implemented a three-dimensional solver for simulating the ultrasound radiation force induced shear wave motion in viscoelastic media using a mapped Chebyshev pseudospectral method. The solver used a body force field that is simulated by the Field II program as the excitation. Then, a velocity-stress formulation was used to calculate the shear wave motion. We used a Voigt model to describe tissue viscoelasticity, and the viscosity is introduced by memory variables and auxiliary differential equations. The top boundary of the domain had a fixed boundary condition to mimic the interaction between an ultrasound probe and the tissue surface. All other sides of the domain were surrounded by perfectly matched layers to minimize the reflections. Time stepping was implemented with the LSODE integrator with variable step sizes. Results show that the waveform of the simulated shear wave is close to experimental observations and equivalent FEM simulations. We used a Fourier-based method to estimate the phase velocities of the shear waves between 100–1000 Hz and a nonlinear regression to estimate the Voigt model parameters. The reconstructions matched within 10% error compared with the true values.

Direct trajectory optimization based on a mapped Chebyshev pseudospectral method

In view of generating optimal trajectories of Bolza problems, standard Chebyshev pseudospectral (PS) method makes the points’ accumulation near the extremities and rarefaction of nodes close to the center of interval, which causes an ill-condition of differentiation matrix and an oscillation of the optimal solution. For improvement upon the difficulties, a mapped Chebyshev pseudospectral method is proposed. A conformal map is applied to Chebyshev points to move the points closer to equidistant nodes. Condition number and spectral radius of differentiation matrices from both methods are presented to show the improvement. Furthermore, the modification keeps the Chebyshev pseudospectral method’s advantage, the spectral convergence rate. Based on three numerical examples, a comparison of the execution time, convergence and accuracy is presented among the standard Chebyshev pseudospectral method, other collocation methods and the proposed one. In one example, the error of results from mapped Chebyshev pseudospectral method is reduced to 5% of that from standard Chebyshev pseudospectral method.

Mapped Chebyshev pseudospectral method for the study of multiple scale phenomena

In the framework of mapped pseudospectral methods, we use a polynomial-type mapping function in order to describe accurately the dynamics of systems developing small size structures. Using error criteria related to the spectral interpolation error, the polynomial-type mapping is compared against previously proposed mappings for the study of collapse and shock wave phenomena. As a physical application, we study the dynamics of two coupled beams, described by coupled nonlinear Schrödinger equations and modeling beam propagation in an atomic coherent media, whose spatial sizes differ up to several orders of magnitude. It is demonstrated, also by numerical simulations, that the accuracy properties of the polynomial-type mapping outperform by orders of magnitude the ones of the other studied mapping functions.