Rosenbrock Method Research Articles

Implicit methods for efficient musculoskeletal simulation and optimal control

The ordinary differential equations for musculoskeletal dynamics are often numerically stiff and highly nonlinear. Consequently, simulations require small time steps, and optimal control problems are slow to solve and have poor convergence. In this paper, we present an implicit formulation of musculoskeletal dynamics, which leads to new numerical methods for simulation and optimal control, with the expectation that we can mitigate some of these problems. A first order Rosenbrock method was developed for solving forward dynamic problems using the implicit formulation. It was used to perform real-time dynamic simulation of a complex shoulder arm system with extreme dynamic stiffness. Simulations had an RMS error of only 0.11 degrees in joint angles when running at real-time speed. For optimal control of musculoskeletal systems, a direct collocation method was developed for implicitly formulated models. The method was applied to predict gait with a prosthetic foot and ankle. Solutions were obtained in well under one hour of computation time and demonstrated how patients may adapt their gait to compensate for limitations of a specific prosthetic limb design. The optimal control method was also applied to a state estimation problem in sports biomechanics, where forces during skiing were estimated from noisy and incomplete kinematic data. Using a full musculoskeletal dynamics model for state estimation had the additional advantage that forward dynamic simulations, could be done with the same implicitly formulated model to simulate injuries and perturbation responses. While these methods are powerful and allow solution of previously intractable problems, there are still considerable numerical challenges, especially related to the convergence of gradient-based solvers.

Automatic symbolic derivation of order conditions for two-stage complex Rosenbrock schemes

Rosenbrock methods are known to be good in numerical modeling of stiff problems. In this work, a new symbolic algorithm is proposed for automating the derivation of order conditions for two-stage Rosenbrock schemes with complex coefficients. Software implementation of the proposed algorithm is discussed in detail.

A new method for simulating the motion of individual ellipsoidal bacteria in microfluidic devices

To successfully perform biological experiments on bacteria in microfluidic devices, control of micron-scale cell motion in the chip-sized environment is essential. Here we describe a new method for simulating the motion of individual bacterial cells in a microfluidic device using a one-way coupling Lagrangian approach combined with rigid body theory. The cell was assumed to be an elastic, solid ellipsoid, and interactions with solid wall boundaries were considered to occur in one of two collision modes, either a "standing" or "lying" collision mode on the surface. The ordinary differential equations were solved along the cell trajectory for the thirteen unknown variables of the translational cell velocity, cell location vector, rotational angular velocity, and four Euler parameters, using the Rosenbrock method based on an adaptive time-stepping technique. As selected applications, we show how this novel simulation method may be applied to the designs of efficient hydrodynamic cell traps in a microfluidic device for bacterial applications and for cell separations. Modeled designs include optimized U-shaped sieve arrays with a single aperture for the hydrodynamic cell trapping, and three kinds of staggered micropillars for cell separations.

A second-order pseudo-transient method for steady-state problems

This article gives a second pseudo-transient method for a special system of nonlinear equations, which arises from chemical reaction rate equations. This method uses a special second-order Rosenbrock method as the discrete difference scheme, which satisfies a linear conservation law. Moreover, it adaptively adjusts the time step in inverse proportion to an arithmetic mean of the current residual and the previous residual at every iteration step. For a singular system of nonlinear equations, under some standard assumptions, local convergence of the new method is addressed. Finally, some promise numerical results are also reported.

High-order [formula omitted]-methods

Implicit methods are the natural choice for solving stiff systems of ODEs. Rosenbrock methods are a class of linear implicit methods for solving such stiff systems of ODEs. In the Rosenbrock methods the exact Jacobian must be evaluated at every step. These evaluations can make the computations costly. By contrast, W -methods use occasional calculations of the Jacobian matrix. This makes the W -methods popular among the class of linear implicit methods for numerical solution of stiff ODEs. However, the design of high-order W -methods is not easy, because as the order of the W -methods increases, the number of order conditions of the W -methods increases very fast. In this paper, we describe an approach to constructing high-order W -methods.

Trees, B-series and exponential integrators

Questions concerning the accuracy of numerical methods for differential equations are often analysed using B-series and other formulations based on rooted trees. The analysis of numerical methods, such as Rosenbrock and certain exponential methods, requires an additional algebraic structure to represent the direct use of Jacobian matrices in the computation. It is shown how this can be done within a context containing a broad review of the existing B-series formulation.

Revisiting the Rosenbrock numerical solutions of the reactor point kinetics equation with numerous examples

The fourth order Rosenbrock method with an automatic step size control feature was described and applied to solve the reactor point kinetics equations. A FORTRAN 90 program was developed to test the computational speed and algorithm accuracy. From the results of various benchmark tests with different types of reactivity insertions, the Rosenbrock method shows high accuracy, high efficiency and stable character of the solution.

Adaptivity in Space and Time for Magnetoquasistatics

This paper addresses fully space-time adaptive magnetic field computations. We describe an adaptive Whitney finite element method for solving the magnetoquasistatic formulation of Maxwell’s equations on unstructured 3D tetrahedral grids. Spatial mesh refinement and coarsening are based on hierarchical error estimators especially designed for combining tetrahedral H(curl)-conforming edge elements in space with linearly implicit Rosenbrock methods in time. An embedding technique is applied to get efficiency in time through variable time steps. Finally, we present numerical results for the magnetic recording write head benchmark problem proposed by the Storage Research Consortium in Japan. Mathematics subject classification: 65M60, 65L06, 78M10

Combining Trust-Region Techniques and Rosenbrock Methods to Compute Stationary Points

Rosenbrock methods are popular for solving a stiff initial-value problem of ordinary differential equations. One advantage is that there is no need to solve a nonlinear equation at every iteration, as compared with other implicit methods such as backward difference formulas or implicit Runge–Kutta methods. In this article, we introduce a trust-region technique to select the time steps of a second-order Rosenbrock method for a special initial-value problem, namely, a gradient system obtained from an unconstrained optimization problem. The technique is different from the local error approach. Both local and global convergence properties of the new method for solving an equilibrium point of the gradient system are addressed. Finally, some promising numerical results are also presented.

Towards a Fully Space-Time Adaptive FEM for Magnetoquasistatics

This paper is concerned with fully space-time adaptive magnetic field computations. We describe a Whitney finite element method for solving the magnetoquasistatic formulation of Maxwell's equations on unstructured 3-D tetrahedral grids. Spatial discretization is done by employing hierarchical tetrahedral -conforming elements proposed by Ainsworth and Coyle. For the time discretization, we use a newly constructed one-step Rosenbrock method ROS3PL with third order accuracy in time. Adaptive mesh refinement and coarsening are based on hierarchical error estimators especially designed for Rosenbrock methods. An embedding technique is applied to get efficiency in time through variable time steps. Finally, we present numerical results for the benchmark problem TEAM 7.

Building a better cell trap: Applying Lagrangian modeling to the design of microfluidic devices for cell biology

In this report, we show how computational fluid dynamics can be applied to the design of efficient hydrodynamic cell traps in microfluidic devices. Modeled hydrodynamic trap designs included a large, multiple-aperture “C-type” sieve for trapping hundreds of cells, flat single-aperture arrays for single cells, and “U-type” hydrodynamic structures with one or two apertures to confine small clusters of cells (∼10–15 cells per trap). Using 3T3 cells as a model system, the motion of each individual cell was calculated using a one-way coupled Lagrangian method. The cell was assumed to be a solid sphere, and interactions with other cells were only considered when a cell sedimented in the trap. The ordinary differential equations were solved along the cell trajectory for the three components of the velocity and location vector by using the Rosenbrock method based on an adaptive time-stepping technique. Validation of the predictive value of modeling, using 3T3 cells flowed through microfluidic devices containing “U-type sieves” under the simulation flow parameters, showed excellent agreement between experiment and simulation with respect to cell number per trap and the uniformity of cell distribution within individual microchambers. For applications such as on-chip cell culture or high-throughput screening of cell populations within a lab-on-a-chip environment, Lagrangian simulations have the potential to greatly simplify the design process.

Novel coupling Rosenbrock‐based algorithms for real‐time dynamic substructure testing

AbstractReal‐time testing with dynamic substructuring is a novel experimental technique capable of assessing the behaviour of structures subjected to dynamic loadings including earthquakes. The technique involves recreating the dynamics of the entire structure by combining an experimental test piece consisting of part of the structure with a numerical model simulating the remainder of the structure. These substructures interact in real time to emulate the behaviour of the entire structure. Time integration is the most versatile method for analysing the general case of linear and non‐linear semi‐discretized equations of motion. In this paper we propose for substructure testing, L‐stable real‐time (LSRT) compatible integrators with two and three stages derived from the Rosenbrock methods. These algorithms are unconditionally stable for uncoupled problems and entail a moderate computational cost for real‐time performance. They can also effectively deal with stiff problems, i.e. complex emulated structures for which solutions can change on a time scale that is very short compared with the interval of time integration, but where the solution of interest changes on a much longer time scale. Stability conditions of the coupled substructures are analysed by means of the zero‐stability approach, and the accuracy of the novel algorithms in the coupled case is assessed in both the unforced and forced conditions. LSRT algorithms are shown to be more competitive than popular Runge–Kutta methods in terms of stability, accuracy and ease of implementation. Numerical simulations and real‐time substructure tests are used to demonstrate the favourable properties of the proposed algorithms. Copyright © 2007 John Wiley & Sons, Ltd.

Adaptive Fuzzy Controller for Efficiency Optimization of Induction Motors

This paper introduces a new technique for efficiency optimization of adjustable-speed drives, with an emphasis on vector-controlled induction motor drives. The technique combines two distinct control methods, namely, online search of the optimal operating point and a model-based efficiency control. For a given operating condition, which is characterized by a given speed and load torque, a search controller (SC); based on the Rosenbrock method) determines the flux level that results in the minimum input power. Once this optimal flux level has been found, this information is utilized to update the rule base of a fuzzy controller that plays the role of an implicit mathematical model of the system. Initially, for any load condition, the rule base yields the rated flux value. As the optimum points associated with the usual operating conditions (given by the required speeds and load torques) are identified by the SC, the rule base is progressively updated such that the fuzzy controller learns to model the optimal operating conditions for the entire torque-speed plane. As the machine parameters are subject to change during operation, the SC is kept active to track possible minor deviation of the optimum point, thus ensuring true optimal efficiency operation.

Line Current Harmonic Elimination and Voltage Control of PWM AC/DC Converter Using a Hybrid Genetic Algorithm

This article presents a novel method for line current harmonic elimination and output voltage control of a PWM AC/DC converter using the principle of hybridization of genetic algorithm (GA) with conventional optimization method. The conventional method selected is the well-known Rosenbrock's (RB) rotating coordinates method. The dual objective of line current harmonic elimination together with output voltage stabilization is redrafted as an optimization task and the switching instances of the PWM converter are then identified using the proposed method. It is observed that while GA offers a near optimal solution to the problem, the method of RB always results in global optima, provided initial values are chosen judiciously. Hence, it is proposed to combine these two methods to gain the advantages of both methods. In such a hybrid optimization method, the task of global search is carried out by GA, while Rosenbrock's method is devoted to local search. Theoretical results are presented and are validated through experimental values. It is shown that fusion of GA with conventional method yields better results.

Treatment of oil-water emulsion by ultrafiltration: A numerical approach

The objective of this work is study of permeate flow variation through a membrane by measuring pressure and concentration of the oil-water emulsion. As a first approach, the tests were made without agitation. For the numerical approach, we developed the cubic finite element method coupled with the Rosenbrock method for the space and temporal discretization of the parabolic partial derivative equations. This method allowed us to solve our equations with boundary conditions of the Fourier type. We proposed and studied various models of permeate flow. The model adopted is that which takes account of the variation of viscosity with level of the polarization layer.

Relaxation of STO-3G and 6-31G* basis set functions in the series of LiF isoelectronic molecules of second row elements

The relaxation effect of functions of STO-3G and 6-31G* basis sets (BS) on their balance is considered in the series of LiF, BeO, BN, and C2 isoelectronic molecules. The values of parameters (exponential factors of basis functions, orbital exponents of Gaussian primitives, and contraction coefficients for the basis functions in molecules are found by the energy minimum criterion from unrestricted Hartree-Fock (UHF) calculations with a direct optimization of parameters: the simplex and Rosenbrock methods. Several optimization schemes differing in the number of parameters varied are performed. An important, from the practical viewpoint, relation between parameters of the basis functions of the sets in question is found through the mean values of Gaussian exponents. The relaxation effect on the variation of the total energy and relative errors in the calculations of interatomic distances, normal vibration frequencies, dissociation energies, and other properties of molecules are examined. The variation of the total energy upon the relaxation of STO-3G and 6-31G* basis functions (RBF) can achieve 1100 kJ/mole and 80 kJ/mole, respectively, and should be taken into account when evaluating energy characteristics, especially for the systems with high polar chemical bonds. The relaxation of STO-3G BS improves the description of molecular properties almost in all cases considered, whereas the relaxation of 6-31G* BS slightly affects its balance.

Determination of optimum operating conditions for industrial dye wastewater treatment using adaptive heuristic criticism pH control

For a pilot-scale application, pH control in the treatment of highly contaminated dye industrial wastewater containing metallic compounds as the main pollutants has been investigated with a method using adaptive heuristic criticism control (AHCC). Subsequent experimentation on between 12 and 18 l of the wastewater was carried out using statistical experimental design methodology to evaluate the effects of three critical factors: slaked lime (calcium hydroxide, Ca(OH) 2) concentration, iron chloride (FeCl 3) concentration and wastewater volume. With these critical factors, the wastewater treatment process is modeled as an appropriate quadratic cost function of the turbidity of the clarified water. The model is optimized with Rosenbrock's method. Response surface topology of the wastewater treatment is given in terms of optimal concentrations of lime water and FeCl 3 and optimal wastewater volume at pH 11.

Reaction mechanism reduction and optimisation for modelling aviation fuel oxidation using standard and hybrid genetic algorithms

This study describes the development of a new binary encoded genetic algorithm for the combinatorial problem of determining a subset of species and their associated reactions that best represent the full starting point reaction mechanism in modelling aviation fuel oxidation. The genetic algorithm has a dual objective in finding a reduced mechanism that best represents aviation fuel oxidation in both a laminar premixed flame and perfectly stirred reactor systems. The number of species in the subset chosen is kept fixed and is specified at the start of the procedure. The genetic algorithm chooses ever improving mechanisms based on an objective function which measures how well the new reduced mechanisms predict a set of species’ profiles simulated by the full mechanism. In order to verify the validity of our approach, a full enumeration was performed on a reduced problem and it was found that the genetic algorithm was able to find the optimum solution to this reduced problem after a few generations. The reduction involved going from 338 reactions involving 67 species to 215 reactions involving 50 species. This corresponded to a 90% CPU time saving in each function evaluation. A second step was to take the reduced reaction mechanism and to use a second real encoded genetic algorithm for the parameter optimisation problem of determining the optimal reaction rate parameters that best model an experimental set of premixed flame and jet stirred reactor species’ profiles. A significant improvement could be seen in the species profiles obtained using the mechanism with the GA optimised rates over those obtained from the original reduced mechanism. Further, in order to increase the efficiency of the second reaction rate coefficient optimisation step, a new hybrid method was developed which incorporates a direct optimisation method (Rosenbrock method) into the GA. A significant improvement in both accuracy and efficiency was apparent in using this new hybrid approach.

Asymptotic stability of Rosenbrock methods for systems of functional differential and functional equations

This paper is concerned with the numerical solution of initial value problems for linear systems of functional differential and functional equations. The adaptation of Rosenbrock methods is considered. It is shown that, under some reasonable assumptions, A-stability Rosenbrock methods can preserve the asymptotic stability of underlying linear systems.

EChem++ – An object-oriented problem solving environment for electrochemistry: Part 4. Adaptive multilevel finite elements applied to electrochemical models Algorithm and benchmark calculations

We present an approach for the modeling and simulation of electrochemical processes based on a general mathematical formulation, for which we expect future extensibility. For the numerical solution we propose the Rothe method with adaptive time integration performed by Rosenbrock schemes. For spatial integration we choose an adaptive finite element algorithm, controlled by a hierarchical a posteriori error estimator. The algorithm is designed to simulate models of potential-controlled transient experiments in 1D cell geometries and diffusional transport. It is tested for cyclic voltammetry with a fast electron transfer under semi-infinite diffusion conditions. Results of selected Rosenbrock methods are compared. We discuss the software design of the algorithm, as well as its coupling to the electrochemical compiler Ecco within an object-oriented framework. The software developed within this study is part of EChem++ and can be obtained freely as open-source code.