Orthogonal Collocation Method Research Articles

Numerical solutions of Schrödinger–Boussinesq system by orthogonal spline collocation method

This paper is concerned with the numerical solutions of Schrödinger–Boussinesq (SBq) system by an orthogonal spline collocation (OSC) discretization in space and Crank–Nicolson (CN) type approximation in time. By using the mathematical induction argument and standard energy method, the proposed CN+OSC scheme is proved to be unconditionally convergent at the order O(τ2+h4) with mesh-size h and time step τ in the discrete L2-norm. We devise a new computation method based on the orthogonal diagonalization techniques (ODT) to realize the proposed CN+OSC scheme. In order to compare the performance of ODT, we devise an alternating direction implicit (ADI) method to compute the CN+OSC scheme for high spatial dimension SBq system. As an alternative implementation, the new method ODT not only exhibits more accurate numerical results, but also demonstrates stronger invariance preserving ability. Numerical results are reported to verify the error estimates and the discrete conservation laws.

Generalized log orthogonal functions spectral collocation method for two dimensional weakly singular Volterra integral equations of the second kind

AbstractIn this article, a generalized log orthogonal functions (GLOFs)‐spectral collocation method to two dimensional weakly singular Volterra integral equations of the second kind is proposed. The mild singularities of the solution at the interval endpoint can be captured by Gauss‐GLOFs quadrature and the shortcoming of the traditional spectral method which cannot well deal with weakly singular Volterra integral equations with limited regular solutions is avoided. A detailed convergence analysis of the numerical solution is carried out. The efficiency of the proposed method is demonstrated by numerical examples.

Unraveling Drug Delivery from Cyclodextrin Polymer-Coated Breast Implants: Integrating a Unidirectional Diffusion Mathematical Model with COMSOL Simulations.

Breast cancer ranks among the most commonly diagnosed cancers worldwide and bears the highest mortality rate. As an integral component of cancer treatment, mastectomy entails the complete removal of the affected breast. Typically, breast reconstruction, involving the use of silicone implants (augmentation mammaplasty), is employed to address the aftermath of mastectomy. To mitigate postoperative risks associated with mammaplasty, such as capsular contracture or bacterial infections, the functionalization of breast implants with coatings of cyclodextrin polymers as drug delivery systems represents an excellent alternative. In this context, our work focuses on the application of a mathematical model for simulating drug release from breast implants coated with cyclodextrin polymers. The proposed model considers a unidirectional diffusion process following Fick's second law, which was solved using the orthogonal collocation method, a numerical technique employed to approximate solutions for ordinary and partial differential equations. We conducted simulations to obtain release profiles for three therapeutic molecules: pirfenidone, used for preventing capsular contracture; rose Bengal, an anticancer agent; and the antimicrobial peptide KR-12. Furthermore, we calculated the diffusion profiles of these drugs through the cyclodextrin polymers, determining parameters related to diffusivity, solute solid-liquid partition coefficients, and the Sherwood number. Finally, integrating these parameters in COMSOL multiphysics simulations, the unidirectional diffusion mathematical model was validated.

An orthogonal spline collocation method for singularly perturbed parabolic reaction–diffusion problems with time delay

An orthogonal spline collocation method for singularly perturbed parabolic reaction–diffusion problems with time delay

Application of the orthogonal collocation method in evaluating the rate of reactions using thermogravimetric analysis data

The orthogonal collocation method offers a solution to the challenge of noise in thermogravimetric analysis data, improving upon the differential methods like the Friedman isoconversional method. This new technique requires only a few data points for interpolation to recreate conversion-time or conversion-temperature curves, from which a smooth reaction rate curve can be derived by differentiating polynomials. Remarkably, this method demands fewer than 20 data points for a precise estimation of reaction rates in single step reactions. The method's validity is supported by both simulated and actual reactions. Users can implement this technique using provided Gnu Octave (a free alternative to MATLAB) codes. Furthermore, the activation energy results of experimental data obtained from the Freidman and orthogonal collocation methods were in good agreement with those evaluated from the integral Vyazovkin and Ortega methods.

Numerical Solution of the Fokker-Planck Equation Using Orthogonal Collocation

This study aims to propose a numerical methodology to solve the Fokker-Planck Equation using the Orthogonal Collocation Method. In this approach, the original problem, represented by a partial differential equation, is rewritten as a system of initial value equations via discretization of spatial variable. The resulting system is integrated considering the Runge-Kutta-Fehlberg Method. The proposed methodology is applied in three case studies that presented an analytical solution. The obtained results demonstrate that the proposed approach is a good alternative for solving this class of problems. Keywords: Fokker-Planck Equation, Orthogonal Collocation, Numerical Method.

BDF2 ADI orthogonal spline collocation method for the fractional integro-differential equations of parabolic type in three dimensions

In this paper, we are concerned with constructing a fast and an efficient alternating direction implicit (ADI) scheme for the fractional parabolic integro-differential equations (FPIDE) with a weakly singular kernel in three dimensions (3D). Our constructed scheme is based on a second-order backward differentiation formula (BDF2) for temporal discretization, orthogonal spline collocation (OSC) method for spatial discretization and a second-order fractional quadrature rule proposed by Lubich for the Riemann-Liouville fractional integral. The stability and convergence of the constructed numerical scheme are derived. Finally, some numerical examples are given to illustrate the accuracy and validity of the BDF2 ADI OSC method. Based on the obtained results, the numerical results are in line with the theoretical ones.

Innovative Optimal Control Path Planning for PMCU in Cruise Ship Construction: A Kinematic and Obstacle Avoidance Model

In cruise ship manufacturing, it is common practice for prefabricated modular cabin units (PMCU) to be assembled in the workshop and transported to a designated installation location on board the ship. This paper proposed a Radau pseudospectral method based on a probabilistic roadmap (PRM) for PMCU path planning in complex deck environments. Firstly, a kinematic model is constructed, and the continuous-time optimal control path planning problem is established. Among them, an area-based obstacle avoidance constraint is applied to the approach. Then, the continuous-time optimal control problem is discretized into a nonlinear programming (NLP) problem using the orthogonal collocation method. For the initial value-sensitive NLP problem solving, a strategy using PRM combined with a heuristic rule is used to obtain a high-quality initial guess. Finally, experiments under two operating scenarios of PMCU pushing demonstrate that the method is capable of generating a better path while ensuring that the whole process is collision-free. The results show that the algorithm proposed in this paper can effectively solve the PMCU pushing path planning problem in the complex deck environment, and the path considering kinematic characteristics can be used for cabin installation, which has strong operability and applicability.

Modeling and Experimental Study of a Water Solar Collector Coupled to Optimized Solar Water Still

This work presented the modeling and the experimental study of a water solar collector coupled with an optimized solar still developed in order to boost fresh water produced in a solar distillation system. The desalination process is currently operated under the climatological conditions of Sfax, Tunisia. To numerically simulate the water solar collector, we developed a dynamic model based on heat and mass transfer from the water solar collector. The obtained set of ordinary differential equations was converted to a set of algebraic system of equations by the functional approximation method of orthogonal collocation. The aim of this study was to present the mathematical model and experimental study of this water solar collector

A Springy Leg and a Double Backflip

This paper presents a simulation study of a planar monopod robot's motion search and control problems. The robot's passively spring-loaded prismatic joint in the lower body (between the foot and the shin) assists it in performing athletic motions. Systems of this kind can balance on a single point, hop, and produce complex movements with a suitable control strategy. We aim to investigate the application of direct orthogonal collocation (DOC) methods to nonlinear motion search problems for obtaining the continuity accuracy of using an ODE solver but exploiting the faster computation and convergence of collocation methods. Moreover, the presented controller allows the robot to get up from the ground, move to the ready position, and replicate the optimized motion (a double backflip) in a dynamic simulation.

Constrained profile estimation for distributed parameter system in one dimension using orthogonal collocation

Several industrial processes fall under the category of Distributed parameter systems (DPS) which are represented as coupled partial differential equations (PDEs). Obtaining accurate and physically meaningful profile estimates of states despite only a limited number of spatially distributed measurements, is a pre-requisite for adequately controlling such systems. In this work, we propose an online constrained state profile estimation approach for DPS. The approach is based on reduced-dimension model represented by differential and algebraic equations (DAE) which in turn is developed by discretization of the PDEs along the spatial coordinate using the orthogonal collocation (OC) method. A profile prediction error based approach is proposed for systematically fixing the dimension of the OC model to be used. Further, to compensate for the inevitable model plant mismatch (MPM) arising in the process of the dimensionality reduction, additive zero mean Gaussian white noise processes are introduced in both state dynamics as well as measurement model. Statistics of these noises are obtained based on simulation based realizations of approximation errors at collocation points and measurement locations. To incorporate the identified correlation between the resulting state and measurement noises in the estimation method, we propose a constrained DAE-EKF (extended Kalman Filter) extension to deal with such correlations. The estimation approach also involves an appropriately formulated optimization based update step to ensure constraint satisfaction over the entire spatial profile leading to physically meaningful state profile estimates. The constrained profile estimator is also integrated with an online moving window based parameter estimation, which enables tracking of both time varying parameters and state profiles. The efficacy of the proposed approach is demonstrated by conducting stochastic simulation studies using two tubular reactor systems. The results show that the proposed constrained online state estimation approach is able to obtain accurate profile estimates at low computational cost, while ensuring that the profile estimates satisfy the given constraints.

Piecewise orthogonal collocation for computing periodic solutions of coupled delay equations

We extend the piecewise orthogonal collocation method to computing periodic solutions of coupled renewal and delay differential equations. Through a rigorous error analysis, we prove convergence of the relevant finite-element method and provide a theoretical estimate of the error. We conclude with some numerical experiments to further support the theoretical results.

An efficient model for capturing gas transients in the energy flow of integrated electricity–gas systems

Gas-fired units provide great flexibility to accommodate the intermittent nature of renewable energy sources and are also considered to be an ideal substitute for retiring coal-fired generations. However, the increasing deployment of gas-fired units deepens interdependency between electric power and natural gas, calling for efficient modeling of the heterogeneous energy flow in the integrated electricity–gas systems (IEGS). Plenty of research works show the necessity of considering gas flow transients in the IEGS, which can be described by a set of partial differential equations (PDEs). To ensure a tractable optimization, the PDEs are often discretized using the finite/infinite difference method, which may result in a large computational scale for achieving good precision. To this end, this paper proposes an efficient model for capturing gas flow transients in the IEGS, which can be taken as the operational constraints in the optimal energy flow model for IEGS coordinated planning, scheduling, and control. For improving the computational efficiency, the PDE-based transient gas flow model is discretized using a space–time orthogonal collocation (OC) method, which provides high-quality results using fewer spatial–temporal discrete points than differential methods. Further, aiming at the difficulty in the determination of the number of space–time collocation points, an adaptive strategy is proposed and embedded in the proposed model utilizing the local fitting errors of state variables related to pipelines. Case studies show the superiority in computational accuracy and efficiency of the proposed model over the existing models based on Wendorff difference and global OC. And an illustrative case based on the modified IEGS benchmark system composed of the IEEE 118-bus power system and the Belgian 20-node gas network exhibits the application value of the proposed model in the optimization and control of the IEGS.

Time Optimal Drag-Based Targeted De-Orbiting for Low Earth Orbit

Controlled de-orbiting plays a crucial role in any space mission to ensure landing of a satellite or capsule in the desired location and preventing damage to people and property on the ground caused by debris. The necessary orbital energy reduction from the initial conditions to the re-entry interface can be achieved using a de-orbit burn or exploiting drag modulation as a control mechanism. The current work perfectly fits in this scenario, proposing a novel algorithm to generate minimum-time optimal trajectories for a satellite ballistic de-orbiting from a Low Earth Orbit (LEO) to the atmospheric re-entry interface. The formulation is written in terms of modified equinoctial orbital parameters, particularly suitable for trajectory analysis and optimization, even in cases of oscillatory problems with large time scales as in the de-orbiting problem. The optimization problem is solved with the MATLAB software GPOPS-II using a hp adaptive Gaussian quadrature orthogonal collocation method. It is formulated as a single-stage optimization problem considering the exposed surface as a control variable. The cost function to be minimized is the final time, while the imposition of an event constraint on the altitude at the de-orbit point ensures its value is in an acceptable range. A novel class of solutions is defined for the algorithm implementation to guarantee the desired values of latitude and longitude. It has been used to generate high-precision optimal trajectories and corresponding control variable laws in different conditions. The identification of a common trend of solutions along an infinite-shaped pattern allowed the possibility to model a wide range of missions, involving different initial conditions and satellites. A subsequent Monte Carlo analysis showed the algorithm validity and robustness with a successful outcome on 500 cases and an error less of 0.5% for most of them.

An ADI orthogonal spline collocation method for a new two-dimensional distributed-order fractional integro-differential equation

An ADI orthogonal spline collocation method for a new two-dimensional distributed-order fractional integro-differential equation

Computational Optimal Impact Angle Control Guidance Laws Weighted by Arbitrary Functions

This paper presents a computational impact angle control guidance law based on the energy cost weighted by arbitrary functions in order to shape the acceleration command as desired. The optimal guidance problem is established in the impact angle frame and is solved by the Gauss orthogonal collocation method. The proposed guidance law is formulated from the generalized optimal control framework, thus a new guidance law that allows achieving a specific guidance goal can be easily obtained in the way of devising a proper weighting function (smooth, piecewise, nonsmooth, or even discontinuous). This property provides more degrees of freedom in the guidance law design to accomplish a specified guidance objective. A hardware experiment is conducted to evaluate the real-time computational capacity of the proposed computational guidance law. Illustrative examples with several types of weighting functions, including smooth, piecewise, nonsmooth, and discontinuous functions, are provided to demonstrate the advantages and capacity of the proposed guidance law.

Nonlinear Model Predictive Path Tracking Control for Autonomous Vehicles Based on Orthogonal Collocation Method

Nonlinear Model Predictive Path Tracking Control for Autonomous Vehicles Based on Orthogonal Collocation Method

Nonlinear Dynamics of Turbine Generator Shaft Trains: Evaluation of Bifurcation Sets Applying Numerical Continuation

Abstract The nonlinear dynamics of turbine generator shaft trains for power generation are investigated in this paper. Realistic models of rotors, pedestals, and nonlinear bearings of partial arc and lemon bore configuration are implemented to compose a nonlinear set of differential equations for autonomous (balanced) and nonautonomous (unbalanced—per ISO) cases. The solution branches of the dynamic system are evaluated with the pseudo-arc length continuation programed by the authors, and the respective limit cycles are evaluated by an orthogonal collocation method and investigated on their stability properties and quality of motion for the respective key design parameters for the rotor dynamic design of such systems, namely, bearing profile and respective pad length, preload and offset, pedestal stiffness and elevation (misalignment), and rotor slenderness. Model order reduction is applied to the finite element rotor model and the reduced system is validated in terms of unbalanced response and stability characteristics. The main conclusion of the current investigation is that the system has the potential to develop instabilities at rotating speeds lower than the threshold speed of instability (evaluated by the linear approach) for specific unbalance magnitude and design properties. Unbalance response (with stable and unstable branches) is evaluated in severely reduced time compared to this applying time integration methods, enabling nonlinear rotor dynamic design of such systems as a standard procedure, and revealing the complete potential of motions (not only local).

Comparative study of different modelling strategies for the dynamic design of cumene reactor in cumene production process

Cumene production is an important intermediate process for the production of phenol and acetone. This paper discusses different modelling strategies to solve the mathematical model of cumene reactor. Five different strategies have been utilised for solving the Partial Differential Equations obtained during the reduction of mathematical model. The schemes include dynamic modelling using Finite Difference Method in OpenModelica, pde solver, FDM in Simulink, MATmol and Orthogonal Collocation Method. A comparative study of the schemes with plant data has been done and it is observed that Orthogonal Collocation Method gives a good agreement with plant data compared to other strategies.

Optimizing energy dissipation in gas foil bearings to eliminate bifurcations of limit cycles in unbalanced rotor systems

High-speed rotor systems mounted on gas foil bearings present bifurcations which change the quality of stability, and may compromise the operability of rotating systems, or increase noise level when amplitude of specific harmonics drastically increases. The paper identifies the dissipating work in the gas film to be the source of self-excited motions driving the rotor whirling close to bearing’s surface. The energy flow among the components of a rotor gas foil bearing system with unbalance is evaluated for various design sets of bump foil properties, rotor stiffness and unbalance magnitude. The paper presents a methodology to retain the dissipating work at positive values during the periodic limit cycle motions caused by unbalance. An optimization technique is embedded in the pseudo-arc length continuation of limit cycles, those evaluated (when exist) utilizing an orthogonal collocation method. The optimization scheme considers the bump foil stiffness and damping as the variables for which bifurcations do not appear in a certain speed range. It is found that secondary Hopf (Neimark–Sacker) bifurcations, which trigger large limit cycle motions, do not exist in the unbalanced rotors when bump foil properties follow the optimization pattern. Period-doubling (flip) bifurcations are possible to occur, without driving the rotor in high response amplitude. Different design sets of rotor stiffness and unbalance magnitude are investigated for the efficiency of the method to eliminate bifurcations. The quality of the optimization pattern allows optimization in real time, and gas foil bearing properties shift values during operation, eliminating bifurcations and allowing operation at higher speed margins. Compliant bump foil is found to enhance the stability of the system.