A modified embedded Runge-Kutta method for cloth simulation

Adaptive time‐stepping methods based on the Monte Carlo Euler method for weak approximation of Itô stochastic differential equations are developed. The main result is new expansions of the computational error, with computable leading‐order term in a posteriori form, based on stochastic flows and discrete dual backward problems. The expansions lead to efficient and accurate computation of error estimates. Adaptive algorithms for either stochastic time steps or deterministic time steps are described. Numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps. Stochastic time steps use Brownian bridges and require more work for a given number of time steps. Deterministic time steps may yield more time steps but require less work; for example, in the limit of vanishing error tolerance, the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps. © 2001 John Wiley & Sons, Inc.

Transient computing scheme named K-MOD has been designed that time-discrete is carried out by implicit Runge-Kutta method, and coarse nodal method is used for neutronics calculation at present. During transient calculation, thermal feedback is introduced to update the cross-section to approximate the transient behavior of the real core based on the method of interpolation on state-points. Corresponding thermal feedback for coupling calculation is programmed and analyzed, and parallel computing is introduced to enhance efficiencies of single-disciplinary codes and data coupling section. Problem of rod insertion and lift in real core of PWR (Pressurized Water Reactor) is tested and compared. The results show that K-MOD is in good agreement with the results of SMART software in both fixed time step and adaptive time step. And the parallel efficiency is also explained.

It is possible to construct fully implicit Runge-Kutta methods like Gaus-Legendre, Radau-IA, Radau-IIA, Lobatto-IIIA, -IIIB, and -IIIC methods of arbitrary high order of convergence. The aim of this paper is to find a new adaptive time stepping for these classes which is based on the embedding technique. Adaptive time step control with embedding is well-known for Runge-Kutta methods, and therefore new embedded methods of order s-1 for the above classes of fully implicit Runge-Kutta methods are constructed. Since these fully implicit methods need the solution of a huge non-linear system of equations different approaches for non-linear equations are discussed and compared. It can be observed that non-linear solvers like the usually used simplified Newton method have a step size restriction if they are applied on higher order methods. We apply our new methods on some lower dimensional ODEs to show that our approach leads to an efficient method.

In this thesis, an efficient rotation-free (RF) triangle is proposed and applied to drape/cloth simulations in which the cloth often under large displacements and rotations. The RF model is a class of thin plate/shell computational models possessing only 3 translational degrees of freedom per director whilst their domains of influence are larger than their domains of integration. An important advantage of RF models is that they do not use rotational degrees of freedom and, thus, are not plagued by the complication in finite rotations. Among the quadrilateral and triangular RF models, the latter possesses no practical restriction on the nodal distribution and appears to be a good candidate for drape/cloth simulations.
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\nThe geometrical linear formulation of the RF model is firstly considered. For straight beams and plates, the curvature is directly obtained through a complete quadratic interpolation of the transverse deflection. For linear curved beams and shells, the curvature change is again derived by the interpolation and the transverse deflection is through projection. The linear RF model is then extended to the geometrical nonlinear analyses by using the corotational framework as well as the small strain and small curvature assumptions. For the RF straight beam and plate, constant tangential bending stiffness matrices which do not need to be updated during the iterative solution process are derived. For the RF curved beam and shell, the bending energies and bending internal forces become a bit complicated. However, the tangential bending stiffness matrices can still be approximated by using the constant matrices as if they are initially straight/flat. The constant approximation exhibits negligible adverse effect on the convergence. Comparing with other exiting RF models, the present RF triangle is simple and physical yet its accuracy is competitive. In its application to static drape simulations, realistic drape configurations with obvious folds are predicted.
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\nThe RF beam is extended to consider static and dynamic analyses of cable structures. Under the same nodal distributions, the present RF model can tolerate larger load increment and time step in static and explicit dynamic analyses, respectively, with respect to the two-node C0beam finite element model.
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\nFor virtual sewing and dynamic cloth simulations, an integrated system is developed by synergizing the RF triangle, explicit time integration, adaptive remeshing, collision handling, human body modeling, sewing forces and a supplementary bending energy to suppress the non-physical sharp fold formation. The predicted steady-state configurations of the garments after sewing appear to be realistic and agree with our daily perception. The predictions for cloth dynamic deformations on human body model also look realistic and natural.
\nThis thesis proposes a simple and efficient rotation-free triangle which is especially suitable for the problems involving large displacements and rotations. Its application in drape/cloth simulations and integration of various techniques in cloth simulations are explored. The present study is of significance in cloth simulations.

The innovation presented is a method for adaptive time-stepping that allows clustering of time steps in portions of the cycle for which flow variables are rapidly changing, based on the concept of using a uniform step in a relevant dependent variable rather than a uniform step in the independent variable time. A user-defined function was developed to adapt the magnitude of the time step (adaptive time step) to a defined rate of change in inlet velocity. Quantitative comparison indicates that the new adaptive time stepping method significantly improves accuracy for simulations using an equivalent number of time steps per cycle.

We present an adaptive time stepping scheme based on the extrapolative method of Barth and Schlick [LN, J. Chem. Phys. 109, 1633 (1998)] to numerically integrate the Langevin equation with a molecular-dynamics potential. This approach allows us to use (on average) a time step for the strong nonbonded force integration corresponding to half the period of the fastest bond oscillation, without compromising the slow degrees of freedom in the problem. We show with simple examples how the dynamic step size stabilizes integration operators, and discuss some of the limitations of such stability. The method introduced uses a slightly more accurate inner integrator than LN to accommodate the larger steps. The adaptive time step approach reproduces temporal features of the bovine pancreatic trypsin inhibitor (BPTI) test system (similar to the one used in the original introduction of LN) compared to short-time integrators, but with energies that are shifted with respect to both LN, and traditional stochastic versions of Verlet. Although the introduction of longer steps has the effect of systematically heating the bonded components of the potential, the temporal fluctuations of the slow degrees of freedom are reproduced accurately. The purpose of this paper is to display a mechanism by which the resonance traditionally associated with using time steps corresponding to half the period of oscillations in molecular dynamics can be avoided. This has theoretical utility in terms of designing numerical integration schemes--the key point is that by factoring a propagator so that time steps are not constant one can recover stability with an overall (average) time step at a resonance frequency. There are, of course, limitations to this approach associated with the complicated, nonlinear nature of the molecular-dynamics (MD) potential (i.e., it is not as straightforward as the linear test problem we use to motivate the method). While the basic notion remains in the full Newtonian problem, it is easier to see the effects when damping is considered to be physical--that is, we do not view our method as a perturbation of Newtonian dynamics, we associate the damping with the environment, for example, a water bath (with gamma approximately 90 ps(-1)) [Zagrovic and Pande, J. Comp. Chem. 24, 1432 (2003)]. All stochastic approaches to MD are stabilized by large physical damping, but here, we are really using it only to show that the resonance frequency can be obtained. Another simplifying assumption used in this paper is "heavy" hydrogen (we take the hydrogen mass to be 10 amu)--the view here is that we are interested primarily in the slowest degrees of freedom, and this approach has effects similar to bond freezing and united atom treatments of hydrogen. So from the point of view of biomolecular applications, the method described here is best suited to studies in which water is not explicit (so that damping in the problem can really be viewed as environmental interaction), and the interest is in slow dynamics where the effects of hydrogen are neglectable. There are a number of parameters in the LN method and the one derived here, and we cannot in a short paper address all adjustments, so our primary goal as a first pass is to show that stability can be recovered for a set of numerically forced (and hence artificial) bond oscillations, and compare stability to fixed-step methods.

In the present study, dynamic stability of a viscoelastic cantilevered pipe conveying fluid which fluctuates harmonically about a mean flow velocity is considered; while the fluid flow is exhausted through an inclined end nozzle. The Euler-Bernoulli beam theory is used to model the pipe and fluid flow effects are modelled as a distributed load along the pipe which contains the inertia, Coriolis, centrifugal and induced pulsating fluid flow forces. Moreover, the end nozzle is modelled as a follower force which couples bending vibrations with torsional ones. The extended Hamilton's principle and the Galerkin method are used to derive the bending-torsional equations of motion. The coupled equations of motion are solved using Runge-Kutta algorithm with adaptive time step and the instability boundary is determined using the Floquet theory. Numerical results present effects of some parameters such as fluid flow fluctuation, bending-to-torsional rigidity ratio, nozzle inclination angle, nozzle mass and viscoelastic material on the stability margin of the system and some conclusions are drawn.

In the previous chapter, we introduced explicit Runge-Kutta (ERK) methods and demonstrated how they can be implemented as a hierarchy of Python classes. For most ODE systems, replacing the simple forward Euler method with a higher-order ERK method will significantly reduce the number of time steps needed to reach a specified accuracy. Furthermore, it often leads to reduced computation time, since the additional cost per time step is outweighed by the reduced number of steps. However, there exists a class of ODEs known as stiff systems, where all the ERK methods require very small time steps, and any attempt to increase the time step leads to spurious oscillations and possible divergence of the solution. Stiff ODE systems pose a challenge for explicit methods, and they are better addressed by implicit solvers such as implicit Runge-Kutta (IRK) methods. IRK methods are well-suited for stiff problems and can offer substantial reductions in computation time when tackling challenging problems.

Molecular Dynamics (MD) simulation is a versatile methodology that has found many applications in materials science, chemistry and biology. In biology, the models employed range from mixed quantum mechanical and fully atomistic to united atom and continuum mechanical. These systems are evolved in discrete time by solving Newton’s equations of motion at each time step. The numerical methods currently in use limit the step size of a typical all atom simulation to 1 femtosecond. This step size limitation means that many steps need to be taken in order to reach biologically relevant time scales. At each time step, an evaluation of the forces on each atom must be performed resulting in heavy computational loads. This work investigates the use of implicit integration methods in MD. Implicit integration methods have been proven superior to their explicit counterparts in classical mechanical simulation, with which MD has many similarities. Longer time steps reduce the number of force evaluations that must be performed and the corresponding computational load. Herein we present results that compare implicit integration techniques with the current standard for molecular dynamics, the explicit velocity Verlet integration scheme. Total energy conservation is used as a metric for evaluating the dependability of simulations in the microcanonical ensemble. In order to understand the nature of the problem, several long simulations were run and analyzed by performing a Fourier analysis on the position, velocity and acceleration signals. Lastly, several methods for improving the viability of implicit integration methods are considered including replacing the Jacobian used in the Quasi-Newton method with a constant, diagonal mass matrix, evaluating the Jacobian infrequently and finding a better prediction of the system configuration to improve the convergence of the Quasi-Newton method.

A procedure for calculating sensitivity coefficients of one-dimensional parabolic mixed initial-boundary value problems is developed. The method is based upon the use of implicit time integration and Newton's method in solving the governing equations. The link between the sensitivity coefficients and the solution method is studied with particular emphasis on the use of adaptive gridding and variable time stepping. The method is employed in the analysis of two model freely propagating premixed laminar flames. The numerical accuracy of the sensitivities is verified and their values are utilized for interpretation of the model results.

Rate optimization of fractional flow reservoir model based on the continuous adjoint method

<p>Natural and induced seismicity is widely investigated, and extensive knowledge has been acquired in the last years, but exact earthquake mechanisms remain elusive and poorly understood. The high impact of earthquakes on human society emphasizes that a deeper understanding of earthquake processes must be a priority in order to improve seismic hazard assessment and mitigate associated risks. Pervasive fluid flow is a key process significantly influencing rock physics and mechanics, and thus has a crucial impact on natural and induced earthquakes. Seismo-hydro-thermo-mechanical (SHTM) modelling is an important nascent branch of geodynamic modelling, which investigates evolution of coupled fluid-solid systems under conditions of both slow tectonic and fast seismic deformation rates. Here, we present a new fully coupled two-dimensional seismo-hydro-mechanical numerical code, with a poro-visco-elasto-plastic rheology, based on fully staggered finite differences with marker-in-cell technique, adaptive time stepping and global Picard iterations. The presented numerical code combines inertial mechanical deformation with pervasive fluid flow. The adaptive time stepping allows the resolution of co-seismic and interseismic phases with time steps ranging from milliseconds to years.</p><p>First, we demonstrate how fluid-bearing subducting rocks are intrinsically seismic and how seismic events in the form of highly localized ruptures spontaneously nucleate along the subduction interface. Nucleation and propagation of such events are driven by rapid fluid pressurization caused by visco-plastic compaction, counterbalanced by a nearly simultaneous and equivalent poroelastic decompaction inside the propagating and rupturing fault. Successive post and interseismic fluid pressure release, generated by poroelastic compaction along the fault, allows strength recovery of the megathrust. The model reproduces the broad range of seismic events present at the subduction interface, including slower events, regular earthquakes, and earthquakes that rupture the entire megathrust and reach velocities on the order of m/s, without employing slip rate dependent frictional properties.</p><p>Next, we show how our approach can be successfully adapted for fluid injection setups to model induced seismicity phenomena. The numerical code successfully modelled fluid induced seismic events and the resulting seismic wave propagation.  Preliminary results show that faults can form spontaneously and grow aseismically at the injection site thereby creating favorable conditions for the development of broad induced seismicity region where different faults can be activated seismically at different time.</p><p>Finally, we outline in short future SHTM modeling directions that should account for fracture-induced dilatation and dynamic permeability variations, thermal effects and three-dimensionality.</p>

Currently, the mechanical dynamics of soft tissue deformation is achieved by numerical time integrations such as the explicit or implicit integration; however, the explicit integration is stable only under a small time step, whereas the implicit integration is computationally expensive in spite of the accommodation of a large time step. This paper presents a cellular neural network method for stable simulation of soft tissue deformation dynamics. The non-rigid motion equation is formulated as a cellular neural network with local connectivity of cells, and thus the dynamics of soft tissue deformation is transformed into the neural dynamics of the cellular neural network. Results show that the proposed method can achieve good accuracy at a small time step. It still remains stable at a large time step, while maintaining the computational efficiency of the explicit integration. The proposed method can achieve stable soft tissue deformation with efficiency of explicit integration for surgical simulation.

Parallel implementation for the two-stage SDIRK methods via diagonalization

Adaptive time-stepping with high-order embedded Runge-Kutta pairs and rejection sampling provides efficient approaches for solving differential equations. While many such methods exist for solving deterministic systems, little progress has been made for stochastic variants. One challenge in developing adaptive methods for stochastic differential equations (SDEs) is the construction of embedded schemes with direct error estimates. We present a new class of embedded stochastic Runge-Kutta (SRK) methods with strong order 1.5 which have a natural embedding of strong order 1.0 methods. This allows for the derivation of an error estimate which requires no additional function evaluations. Next we derive a general method to reject the time steps without losing information about the future Brownian path termed Rejection Sampling with Memory (RSwM). This method utilizes a stack data structure to do rejection sampling, costing only a few floating point calculations. We show numerically that the methods generate statistically-correct and tolerance-controlled solutions. Lastly, we show that this form of adaptivity can be applied to systems of equations, and demonstrate that it solves a stiff biological model 12.28x faster than common fixed timestep algorithms. Our approach only requires the solution to a bridging problem and thus lends itself to natural generalizations beyond SDEs.