Adaptive Collocation Method Research Articles

An adaptive collocation method on implicit domains using weighted extended THB-splines

Implicit representations possess many merits when dealing with geometries with certain properties, such as small holes, reentrant corners and other complex details. Truncated hierarchical B-splines (THB-splines) has recently emerged as a novel tool in many fields including design and analysis due to its local refinement ability. In this paper, we propose an adaptive collocation method with weighted extended THB-splines (WETHB-splines) on implicit domains. We modify the classification strategy for the WETHB-basis, and the centers of the supports of inner THB-splines on each level are chosen to be collocation points. We also use weighted collocation in the transition regions, in order to enrich information concerning the hierarchical basis. In contrast to the traditional WEB-collocation method, the proposed approach possesses much higher convergence rate. To show the efficiency and superiority of the proposed method, numerical examples in two and three dimensions are performed to solve Poisson's equations.

On the Convergence of Adaptive Stochastic Collocation for Elliptic Partial Differential Equations with Affine Diffusion

Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting. Extensions to other variants of adaptive collocation methods (including the classical one proposed in the paper Dimension-adaptive tensor-product quadratuture Computing (2003) by T. Gerstner and M. Griebel) is explored.

Mission reconstruction for launch vehicles under thrust drop faults based on deep neural networks with asymmetric loss functions

Launch failures of launch vehicles due to thrust drop faults have occurred throughout aerospace history, thus studying Mission Reconstruction (MRC) problems is crucial to alleviate the losses resulting from the thrust drop faults. Unlike the traditional trajectory optimization problem with a specified target orbit, the rescue orbit and flight trajectory in the MRC problems need to be optimized simultaneously. To solve the MRC problems rapidly and accurately, a deep neural network-based adaptive collocation method (DNN-based ACM) is proposed. In the offline part, using the datasets generated by convex optimization and adaptive collocation method, deep neural networks (DNN) are trained to map the relationship from the fault states to the optimal rescue orbital elements and terminal control variables. During the DNNs training, an asymmetric loss function is proposed to avoid the positive error of the semi-major axis which renders the trajectory optimization unsolvable. In the online application of the DNNs, the optimal rescue orbit is determined and proper initial guesses are set for the MRC problems. The flight trajectory from the fault position to the optimal rescue orbit can be effectively solved by the ACM with the proper initial guesses. Simulation results validate the effectiveness of the DNN-based ACM, and substantiate the high efficiency performance and optimal performance of the presented MRC method.

Powered soft landing guidance method for launchers with non-cluster configured engines

This paper proposes a guidance method for the powered soft landing of a launcher with non-cluster configured engines, for which it is difficult to maintain a low thrust-to-weight ratio (TWR) preferable for rocket recovery. For rockets in service with one or two engines rather than a cluster of engines in the boosters, it is a challenge to deeply throttle the engines to a low level, especially lower than 30%, and if one of the two engines is shut down to reduce the thrust, the attitude control is also difficult due to thrust asymmetry. The theoretical analysis and simulations show that the higher the TWR is, the smaller the feasible region for a soft landing becomes. The analysis also reveals that the bang–bang solutions of fuel-optimal methods are vulnerable to uncertainties and disturbances. A thrust regulation rate constraint is proposed or convexified to reconstruct the landing problem under high TWRs. An initial guess based on mean thrust and relaxing the initial velocity constraint is proposed, making both the adaptive collocation method and the successive convex programming applicable for real-time landing planning. Simulations under TWRs larger than four show that the proposed methods ensure a safe landing and are more adaptive than fuel-optimal solutions when facing disturbances and uncertainties.

An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs

In this paper, we introduce two new methods to solve systems of ordinary differential equations. The first method is constituted of the generalized Bernstein functions, which are obtained by Bernstein polynomials, and operational matrix of differentiation with collocation method. The second method depends on tau method, the generalized Bernstein functions and operational matrix of differentiation. These methods produce a series which is obtained by non-polynomial functions set. We give the standard Bernstein polynomials to explain the generalizations for both methods. By applying the residual correction procedure to the methods, one can estimate the absolute errors for both methods and may obtain more accurate results. We apply the methods to some test examples including linear system, non-homogeneous linear system, nonlinear stiff systems, non-homogeneous nonlinear system and chaotic Genesio system. The numerical shows that the methods are efficient and work well. Increasing m yields a decrease on the errors for all methods. One can estimate the errors by using the residual correction procedure.

Analysis of an adaptive collocation solution for retarded and neutral delay systems

This paper introduces an adaptive collocation method to solve retarded and neutral delay differential equations (RDDEs and NDDEs) with constant or time-dependent delays. The delays are allowed to be small or become vanishing during the integration. We determine the convergence properties of the proposed method for neutral equations with solutions in appropriate Sobolev spaces. It is shown that the proposed scheme enjoys the spectral accuracy. Numerical results show that the proposed method can be implemented in an efficient and accurate manner for a wide range of RDDE and NDDE model problems.

An Adaptive Collocation Method with Weighted Extended PHT-Splines

This paper presents an adaptive collocation method with weighted extended PHT-splines. The authors modify the classification rules for basis functions based on the relation between the basis vertices and the computational domain. The Gaussian points are chosen to be collocation points since PHT-splines are C1 continuous. The authors also provide relocation techniques to resolve the mismatch problem between the number of basis functions and the number of interpolation conditions. Compared to the traditional Greville collocation method, the new approach has improved accuracy with fewer oscillations. Several numerical examples are also provided to test our the proposed approach.

An adaptive collocation method for structural fuzzy uncertainty analysis

PurposeConsidering that uncertain factors widely exist in engineering practice, an adaptive collocation method (ACM) is developed for the structural fuzzy uncertainty analysis.Design/methodology/approachACM arranges points in the axis of the membership adaptively. Through the adaptive collocation procedure, ACM can arrange more points in the axis of the membership where the membership function changes sharply and fewer points in the axis of the membership where the membership function changes slowly. At each point arranged in the axis of the membership, the level-cut strategy is used to obtain the cut-level interval of the uncertain variables; besides, the vertex method and the Chebyshev interval uncertainty analysis method are used to conduct the cut-level interval uncertainty analysis.FindingsThe proposed ACM has a high accuracy without too much additional computational efforts.Originality/valueA novel ACM is developed for the structural fuzzy uncertainty analysis.

Wavelet Methods and Adaptive Grids in One-Dimensional Movable Boundary Problems

Adaptive wavelet collocation methods use wavelet transform and filtering to generate adaptive grids. However, if the boundary moves, the grid becomes aberrant. It baffles wavelet transform and makes the adaptive wavelet methods lose advantages on computational efficiency. This paper develops a series of methods or skills to effectively perform wavelet transform and to generate adaptive grids for one-dimensional movable boundary problems. The methods remain the original inner grid points and keep the grid in the original-nested structure, in order to remain scanty during the whole computing process. For boundary extending, the adaptive wavelet program begins to run on the very new grid beyond the original boundary once it reaches a nested structure, which is called the Intermittent Adaptive Method as a consequence. If the boundary extends tremendously, the new nested grids can be combined to a greater nested grid for further efficiency, which is named the Grid Combine Method. While for boundary contracting, a fictitious boundary is addressed to replace the original boundary that will recede, so wavelet transform can be successfully performed on the original nested grid. Finally, two numerical tests, local features moving and gas gun, were resolved and discussed to show the evolution process of the adaptive grids with the boundaries moving. For boundary contracting, the valid points decrease because the computation domain recedes; while for boundary extending, the valid point numbers vary between a range that almost remains unchanged.

An Adaptive Collocation Method for Solving Delay Fractional Differential Equations

In this article, an adaptive collocation method is investigated for solving delay fractional differential equations (DFDEs). The fractional derivative is described in the Caputo–Fabrizio sense, that is a new fractional derivative with non-singular kernel. This new definition has more advantages over the definition of Caputo fractional derivative that we consider in our numerical method. Our technique is based upon an adaptive pseudospectral method. First, we divide the interval of the problem into a uniform mesh and consider the Legendre polynomials on each subinterval then using the Chebysheve collocation points the given DFDE reduces to a system of algebraic equations. One of the reasons for using the adaptive methods is their superiority in solving the problem containing delay terms. The technique is simple to implement and yields precise results. The error approximation and convergence properties of the method are discussed. The proposed method in this investigation is easy and effective for solving DFDEs and can provide an accuracy approximate solution.

Autonomous trajectory planning for launch vehicle under thrust drop failure

An online autonomous rescue strategy and the algorithm for launch vehicle are studied in the case of a thrust drop during ascending flight. First, an iterative guidance method and numerical integration are used to evaluate the remaining carrying capacity. When the target orbit is unreachable, but the lowest safe orbit could be met, an optimal rescue orbit (ORO) needs to be solved. Using the geocentric angle estimation, orbit coordinate system transformation, and convexification, the maximum-height circular orbit in the orbit plane determined by states at failure time, is found under the constraint of geocentric angle. Then, an optimal circular orbit (OCO) relaxing the above constraint is solved by the adaptive collocation method (ACM) using the former solution as initial values. Finally, a decision whether to adjust other orbital elements by comparing the height between OCO and target orbit is made. The algorithm gives priority to the orbit height, optimizes the ORO based on the objective function weights, and makes full use of the advantages of geocentric angle estimation: simplifying the terminal constraints, making the convex optimization achieve good convergence, and improving the effectiveness of the ACM under reasonable initial guess. The simulation results show that the proposed strategy and algorithm are adaptable and, convergent for onboard application.

Sequential optimization and fuzzy reliability analysis for multidisciplinary systems

To meet the rising demand for high reliability in complex multidisciplinary engineering systems, more attention has been paid to reliability-based multidisciplinary design optimization (RBMDO). In this paper, a sequential optimization and fuzzy reliability analysis (SOFRA) method for multidisciplinary systems is developed to decouple the fuzzy reliability analysis from the optimization. In SOFRA, the multidisciplinary design optimization (MDO) and fuzzy reliability analysis are conducted in a sequential manner. Furthermore, a novel adaptive collocation method (ACM) is proposed to conduct the fuzzy reliability analysis for multidisciplinary systems. The ACM arranges points adaptively at the axis of the membership to obtain more accurate results. The shifting distance of the constraint is calculated by the bi-section method. Both numerical and engineering examples are used to demonstrate the validity of the proposed method.

Extended BACOLI

BACOLI is a Fortran software package for solving one-dimensional parabolic partial differential equations (PDEs) with separated boundary conditions by B-spline adaptive collocation methods. A distinguishing feature of BACOLI is its ability to estimate and control error and correspondingly adapt meshes in both space and time. Many models of scientific interest, however, can be formulated as multiscale parabolic PDE systems, that is, models that couple a system of parabolic PDEs describing dynamics on a global scale with a system of ordinary differential equations describing dynamics on a local scale. This article describes the Fortran software eBACOLI, the extension of BACOLI to solve such multiscale models. The performance of the extended software is demonstrated to be statistically equivalent to the original for purely parabolic PDE systems. Results from eBACOLI are given for various multiscale models from the extended problem class considered.

An Adaptive Spectral Parametric Method for Solving Nonlinear Initial Value Problems

An adaptive spectral collocation method based on the parametric iteration method is established for simulating the solution of nonlinear initial value problems of ordinary differential equations. The implementation of the presented approach is demonstrated by solving several nonlinear problems. The numerical results obtained using the developed method are compared with those of the Matlab ode45 and ode87 solvers. This comparison shows that the developed algorithm is a promising tool to handle these kinds of problems.

An adaptive simulation of nonlinear heat and moisture transfer as a boundary value problem

This work presents an alternative view on the numerical simulation of diffusion processes applied to the heat and moisture transfer through porous building materials. Traditionally, by using the finite-difference approach, the discretization follows the Method Of Lines (MOL), when the problem is first discretized in space to obtain a large system of coupled Ordinary Differential Equations (ODEs). Thus, this paper proposes to change this viewpoint. First, we discretize in time to obtain a small system of coupled ODEs, which means instead of having a Cauchy (Initial Value) Problem (IVP), we have a Boundary Value Problem (BVP). Fortunately, BVPs can be solved efficiently today using adaptive collocation methods of high order. To demonstrate the benefits of this new approach, three case studies are presented, in which one of them is compared with experimental data. The first one considers nonlinear heat and moisture transfer through one material layer while the second one considers two material layers. Results show how the nonlinearities and the interface between materials are easily treated, by reasonably using a fourth-order adaptive method. Finally, the last case study compares numerical results with experimental measurements, showing a good agreement.

An Adaptive WENO Collocation Method for Differential Equations with Random Coefficients

The stochastic collocation method for solving differential equations with random inputs has gained lots of popularity in many applications, since such a scheme exhibits exponential convergence with smooth solutions in the random space. However, in some circumstance the solutions do not fulfill the smoothness requirement; thus a direct application of the method will cause poor performance and slow convergence rate due to the well known Gibbs phenomenon. To address the issue, we propose an adaptive high-order multi-element stochastic collocation scheme by incorporating a WENO (Weighted Essentially non-oscillatory) interpolation procedure and an adaptive mesh refinement (AMR) strategy. The proposed multi-element stochastic collocation scheme requires only repetitive runs of an existing deterministic solver at each interpolation point, similar to the Monte Carlo method. Furthermore, the scheme takes advantage of robustness and the high-order nature of the WENO interpolation procedure, and efficacy and efficiency of the AMR strategy. When the proposed scheme is applied to stochastic problems with non-smooth solutions, the Gibbs phenomenon is mitigated by the WENO methodology in the random space, and the errors around discontinuities in the stochastic space are significantly reduced by the AMR strategy. The numerical experiments for some benchmark stochastic problems, such as the Kraichnan-Orszag problem and Burgers’ equation with random initial conditions, demonstrate the reliability, efficiency and efficacy of the proposed scheme.

An Adaptive Multi-Element Probabilistic Collocation Method for Statistical EMC/EMI Characterization

An adaptive multi-element probabilistic collocation (ME-PC) method for quantifying uncertainties in electromagnetic compatibility and interference phenomena involving electrically large, multi-scale, and complex platforms is presented. The method permits the efficient and accurate statistical characterization of observables (i.e., quantities of interest such as coupled voltages) that potentially vary rapidly and/or are discontinuous in the random variables (i.e., parameters that characterize uncertainty in a system's geometry, configuration, or excitation). The method achieves its efficiency and accuracy by recursively and adaptively dividing the domain of the random variables into subdomains using as a guide the decay rate of relative error in a polynomial chaos expansion of the observables. While constructing local polynomial expansions on each subdomain, a fast integral-equation-based deterministic field-cable-circuit simulator is used to compute the observable values at the collocation/integration points determined by the adaptive ME-PC scheme. The adaptive ME-PC scheme requires far fewer (computationally costly) deterministic simulations than traditional polynomial chaos collocation and Monte Carlo methods for computing averages, standard deviations, and probability density functions of rapidly varying observables. The efficiency and accuracy of the method are demonstrated via its applications to the statistical characterization of voltages in shielded/unshielded microwave amplifiers and magnetic fields induced on car tire pressure sensors.

Solution of time-varying delay systems using an adaptive collocation method

An adaptive Legendre–Gauss–Radau collocation method is presented for solving time-varying delay systems. By dividing the domain of the problem into a uniform mesh, the delay system is replaced with a sequence of initial value problems. The method developed in this paper adjusts both the mesh spacing and the degree of the polynomial on each mesh interval in order a specified error tolerance is satisfied. We investigate some error estimates on the whole domain of interest. Some numerical examples are given and comparisons with other methods are made to demonstrate the efficiency and accuracy of the proposed method.

Stochastic collocation with kernel density estimation

The stochastic collocation method has recently received much attention for solving partial differential equations posed with uncertainty, i.e., where coefficients in the differential operator, boundary terms or right-hand sides are random fields. Recent work has led to the formulation of an adaptive collocation method that is capable of accurately approximating functions with discontinuities and steep gradients. These methods, however, usually depend on an assumption that the random variables involved in expressing the uncertainty are independent with marginal probability distributions that are known explicitly. In this work we combine the adaptive collocation technique with kernel density estimation to approximate the statistics of the solution when the joint distribution of the random variables is unknown.

A Twofold Spline Chebyshev Linearization Approach for a Class of Singular Second-Order Nonlinear Differential Equations

The aim of this paper is to present a twofold approach for the numerical solution of a class of singular second-order nonlinear differential equations. The first is based on a modified version of an adaptive spline collocation method (ASCM). The second is a patching approach (PASCM) that splits the problem domain into two subintervals: Chebyshev economization procedure is implemented in the vicinity of the singular point and outside this domain the resulting initial or boundary value problem is handled by the (ASCM). The second strategy is based on the linearization of the nonlinear term about the given initial condition at the singular point. The choice of either technique relies on the specified boundary or initial conditions. Performance of the approach is investigated numerically through a number of application examples that demonstrate the efficiency of the approach and that it has O(h 4) rate of convergence. Results confirm that the scheme yields highly accurate results when compared with the exact and/or numerical solutions that exist in the literature.