a probabilistic collocation eulerian lagrangian localized adjoint method on sparse grids for assessing co2 leakage through wells in randomly heterogeneous porous media abstract we develop a probabilistic collocation eulerian lagrangian localized adjoint method on sparse grids for assessing co2 leakage through wells in randomly heterogeneous porous media by utilizing the intrinsic mathematical numerical and physical properties of the mathematical model we model the process in which co2 is injected into a deep aquifer spreads within the aquifer and upon reaching a leaky well rises up to a shallower aquifer to quantify the leakage rate which depends on the pressure build up in the aquifer due to injection and the buoyancy of co2 the underlying eulerian lagrangian framework has high potential to improve the efficiency and accuracy for the numerical simulation of complex flow and transport processes in co2 sequestration the sparse grid probabilistic collocation framework adds computationally efficient uncertainty quantification functionality onto pre existing eulerian lagrangian methods in a nonintrusive manner it also provides a scalable framework to consider uncertainty in a straightforward parallel manner preliminary numerical experiments show the feasibility and potential of the method

shadow hybrid monte carlo an efficient propagator in phase space of macromolecules shadow hybrid monte carlo shmc is a new method for sampling the phase space of large molecules particularly biological molecules it improves sampling of hybrid monte carlo hmc by allowing larger time steps and system sizes in the molecular dynamics md step the acceptance rate of hmc decreases exponentially with increasing system size n or time step t this is due to discretization errors introduced by the numerical integrator shmc achieves an asymptotic o n1 4 speedup over hmc by sampling from all of phase space using high order approximations to a shadow or modified hamiltonian exactly integrated by a symplectic md integrator shmc satisfies microscopic reversibility and is a rigorous sampling method shmc requires extra storage modest computational overhead and a reweighting step to obtain averages from the canonical ensemble this is validated by numerical experiments that compute observables for different molecules ranging from a small n alkane butane with four united atoms to a larger solvated protein with 14 281 atoms in these experiments shmc achieves an order magnitude speedup in sampling efficiency for medium sized proteins sampling efficiency is measured by monitoring the rate at which different conformations of the molecules dihedral angles are visited and by computing ergodic measures of some observables

on the parametrization of finite rotations in computational mechanics a classification of concepts with application to smooth shells this paper concerns the computational treatment of large rotations with application to the finite element discretisation of smooth shells formulations based on various rotational parametrizations are reviewed and classified with respect to their update structure category ii is based on total rotational degrees of freedom leading to an additive update structure whereas category i relies on linearized rotational degrees of freedom leading to a multiplicative update structure based on this classification new formulations are developed among them are the rodrigues formula applied within category ii with additive update structure for the two components of the rotation vector and two successive elementary rotations in the sense of euler angles applied within category i yielding a singularity free formulation the numerical examples confirm that every rotational parametrization when applied within category ii leads to singularities whereas the application within category i enables the calculation of overall rotations unrestricted in size without any singularity

critical study of newmark scheme on manifold of finite rotations in this paper we show that the usual material form of the newmark time stepping scheme of finite rotations is only a simplified version of the correct formula this is because spin material rotation vectors angular velocity vectors and angular accelerations vectors belong to different tangential vector spaces of a manifold at separate time moments we give corrected newmark time stepping schemes for material description and for spatial description

velocity based approach in non linear dynamics of three dimensional beams with enforced kinematic compatibility abstract in the paper we present a new finite element formulation for the dynamic analysis of geometrically exact three dimensional beams we limit our studies to implicit time integration schemes and possible approaches for increasing their robustness and numerical stability in contrast to standard displacement rotation based approach we present here a spatial and temporal discretization based on velocities and angular velocities to describe the rotational degrees of freedom quaternions are used the time integration scheme and the governing equations of the three dimensional beam are modified accordingly in the numerical implementation the galerkin type discretization is employed to obtain the finite element formulation of the problem the result of our studies is simple but accurate efficient and robust numerical model

statically and kinematically exact nonlinear theory of rods and its numerical verification abstract the entirely general nonlinear theory of rods is developed two vector equilibrium equations valid on the rod axis are derived exactly by direct integration of the mechanical balance laws of continuum mechanics over the rod cross section then the energetically conjugate exact 6 field kinematics of the rod axis is implied in an integral identity which expresses the principle of the virtual work the kinematics consists of the displacement vector of the axis describing the average translation and the proper orthogonal tensor representing the average rotation of the cross section the integral identity also suggests the required general form of the constitutive relations they are already approximate from the physical side are then the best place to make geometric approximations and abbreviations to a certain constrained theory the constitutive relations the only nonexact segment of the theory are treated here as an interchangeable element supplementing the exactly formulated principle of virtual work the solution of the nonlinear rod problem becomes nonstandard here for the configuration space is the nonlinear manifold containing the rotation group so 3 applying the finite element method the proper parameterization linearization interpolation accumulation and discretization procedures in so 3 group are discussed a library of 2 3 4 node 1 d isoparametric elements is developed and the computer program ram is worked out the presented numerical tests contain several problems of highly nonlinear plane and spatial deformations of rods with constitutive relations taken at the moment from the classical isotropic elastic beams in all the tests the developed elements in comparison with the solutions known in the literature show a good performance

erratum to a comparison of generalized hybrid monte carlo methods with and without momentum flip j comput phys 228 2009 2256 2265 the generalized hybrid monte carlo ghmc method combines metropolis corrected constant energy simulations with a partial random refreshment step in the particle momenta the standard detailed balance condition requires that momenta are negated upon rejection of a molecular dynamics proposal step the implication is a trajectory reversal upon rejection which is undesirable when interpreting ghmc as thermostated molecular dynamics we show that a modified detailed balance condition can be used to implement ghmc without momentum flips the same modification can be applied to the generalized shadow hybrid monte carlo gshmc method numerical results indicate that ghmc gshmc implementations with momentum flip display a favorable behavior in terms of sampling efficiency i e the traditional ghmc gshmc implementations with momentum flip got the advantage of a higher acceptance rate and faster decorrelation of monte carlo samples the difference is more pronounced for ghmc we also numerically investigate the behavior of the ghmc method as a langevin type thermostat we find that the ghmc method without momentum flip interferes less with the underlying stochastic molecular dynamics in terms of autocorrelation functions and it to be preferred over the ghmc method with momentum flip the same finding applies to gshmc

large rotations revisited application of lie algebra abstract the paper presents an elementary introduction and application of lie algebra to compound groups of large rotations following an elementary overview of the foundations of lie algebra the authors discuss the rotational groups lo 3 and the exponential form of the rotation matrix these yield a practical link between the elegance of lie group theory and engineering kinematics some illustrative examples as the standard stanford manipulator indicate interesting possibilities on enlarged applications concerning robotic mechanisms vehicles and biosystems

a two grid eulerian lagrangian localized adjoint method to miscible displacement problems with dispersion term abstract in this paper we propose a novel two grid method of eulerian lagrangian localized adjoint method ellam mixed finite element method mfem solutions for nonlinear and coupled miscible displacement problems which describe complex fluid flow processes in porous media an ellam is used to solve the nonlinear convection diffusion equation for concentration and we present a mfem to compute the pressure equation for the pressure and darcy velocity a two grid scheme is investigated to linearize and decouple the mixed method equations it is illustrated that the coarse space can be selected as coarse as h o h 1 2 and the asymptotically optimal approximation as the nonlinear schemes can be still achieved as a result a large class of such nonlinear and coupled problems can be solved efficiently with large time steps

a simplified implicit newmark integration scheme for finite rotations implicit newmark schemes for integration of finite rotations in structural and continuum mechanics typically are more complicated than those used for translational motion using euler s representation of the rotation tensor l in terms of a rotation vector f numerical integration connecting the values f n l n and f n 1 l n 1 at the beginning and end of the time step is expressed in terms of an incremental rotation vector q and the associated incremental rotation tensor l q here it is shown using backward differentiation that neglecting third order terms in q the approximation for the angular velocity w in terms of q and its time derivative has the same form as that between velocity and displacement consequently the simplified newmark scheme in terms of q w email protected has the same form as that for updating translations details of the newmark scheme and an analytical expression for the tangent stiffness tensor for the associated newton raphson iteration procedure have been presented for rigid body dynamics the resulting integration scheme has been tested on a nontrivial problem of three dimensional motion of a rigid body using a constant time step the results justify the use of the simplified newmark scheme for finite rotations

matrix tensor notation part ii skew and curved coordinates in part i a notation called matrix tensor notation was introduced for rectilinear orthogonal coordinates part ii discusses how the same notation is equally efficient for vectors and 2nd order tensors in skew bases and skew curved coordinates without considering coordinates first vectors and 2nd order tensors are described in skew bases covariant and contravariant descriptions being distinguished similar to tensor notation a consistent interpretation of deformation and strain in terms of skew bases is given the use of the transpose symbol complicates the matrix algebra but integrates it with tensor algebra which makes it possible to interpret customary tensor equations as relations in space with all the advantages that an isomorphism with euclidean space has it is demonstrated how different metrics can be assigned arbitrarily to define higher dimensional orthogonal vectors in abstract higher dimensional space a consistent notation is given to distinguish between subspace and subbase to write higher order tensors as matrices or 2nd order tensors as vectors partial transpose is introduced which is a formal stretching operation to write any higher order tensor product operation as matrix vector product the description of the variable bases in generally curved coordinates is given in this notation together with the corresponding notation for the partial differentiation rules the vector operations grad div and curl are discussed and an almost physical notation for the christoffel symbols is given however the replacement of tensor notation for higher than second order tensors by matrix tensor notation is not feasible a version of matrix tensor notation that merges with the customary printed form is presented in an appendix further applications are deferred to part iii applying the new notation to curved coordinates in function space and the equations of mechanics projected on finite dimensional subspaces

long run accuracy of variational integrators in the stochastic context this paper presents a lie trotter splitting for inertial langevin equations geometric langevin algorithm and analyzes its long time statistical properties the splitting is defined as a composition of a variational integrator with an ornstein uhlenbeck flow assuming that the exact solution and the splitting are geometrically ergodic the paper proves the discrete invariant measure of the splitting approximates the invariant measure of inertial langevin equations to within the accuracy of the variational integrator in representing the hamiltonian in particular if the variational integrator admits no energy error then the method samples the invariant measure of inertial langevin equations without error numerical validation is provided using explicit variational integrators with first second and fourth order accuracy

a patch that imparts unconditional stability to explicit integrators for langevin like equations this paper extends the results in 8 to stochastic differential equations sdes arising in molecular dynamics it implements a patch to explicit integrators that consists of a metropolis hastings step the patched integrator preserves the sde s equilibrium distribution and is accurate on finite time intervals as a corollary this paper proves the integrator s accuracy in estimating finite time dynamics along an infinitely long solution a first in molecular dynamics the paper also covers multiple time steps holonomic constraints and scalability finally the paper provides numerical tests supporting the theory

the quaternion based three dimensional beam theory this paper presents the equations for the implementation of rotational quaternions in the geometrically exact three dimensional beam theory a new finite element formulation is proposed in which the rotational quaternions are used for parametrization of rotations along the length of the beam the formulation also satisfies the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal in its weak form a strict use of the quaternion algebra in the derivation of governing equations and for the numerical solution is presented several numerical examples demonstrate the validity performance and accuracy of the proposed approach c 2009 elsevier b v all rights reserved

a new look into the kinematics and dynamics of finite rigid body rotations using lie group theory abstract a systematic theoretical approach is presented in an effort to provide a complete and illuminating study on kinematics and dynamics of rigid bodies rotating about a fixed point specifically this approach is based on some fundamental concepts of differential geometry with particular reference to lie group theory this treatment is motivated by the form of the configuration space corresponding to large rigid body rotation which is a differentiable manifold possessing group properties first the basic steps of the classical approach on the subject are briefly summarized then some geometrical tools are presented which are essential for supporting and illustrating the steps and findings of the new approach finally the emphasis is placed on a thorough investigation of the problem of finite rotations a key idea is the introduction of a canonical connection matching the manifold and group properties of the configuration space this proves to be sufficient and effective for performing the kinematics next following the selection of an appropriate metric the dynamics is also carried over the present approach is theoretically more demanding than the traditional treatments in engineering but brings substantial benefits in particular an elegant interpretation is provided for all the quantities with fundamental importance in both rigid body kinematics and dynamics most importantly this also leads to a correction of some misconceptions and geometrical inconsistencies in the field among other things the deeper understanding of the theoretical concepts provides powerful insight and a strong basis for the development of efficient numerical techniques in problems of solid and structural mechanics involving large rotations

global transformations of nonlinear systems recent results have established necessary and sufficient conditions for a nonlinear system of the form dot x t f x t u t g x t with f 0 0 to be locally equivalent in a neighborhood of the origin in rnto a controllable linear system we combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system in doing so we introduce a technique for constructing a transformation under the assumptions that g ldot f dotg ad n 1 f ldotg span an n dimensional space and that g ldot f ldot g ad n 2 f ldotg is an involutive set

multiple time stepping generalized hybrid monte carlo methods performance of the generalized shadow hybrid monte carlo gshmc method 1 which proved to be superior in sampling efficiency over its predecessors 2 4 molecular dynamics and hybrid monte carlo can be further improved by combining it with multi time stepping mts and mollification of slow forces we demonstrate that the comparatively simple modifications of the method not only lead to better performance of gshmc itself but also allow for beating the best performed methods which use the similar force splitting schemes in addition we show that the same ideas can be successfully applied to the conventional generalized hybrid monte carlo method ghmc the resulting methods mts ghmc and mts gshmc provide accurate reproduction of thermodynamic and dynamical properties exact temperature control during simulation and computational robustness and efficiency mts ghmc uses a generalized momentum update to achieve weak stochastic stabilization to the molecular dynamics md integrator mts gshmc adds the use of a shadow modified hamiltonian to filter the md trajectories in the hmc scheme we introduce a new shadow hamiltonian formulation adapted to force splitting methods the use of such hamiltonians improves the acceptance rate of trajectories and has a strong impact on the sampling efficiency of the method both methods were implemented in the open source md package protomol and were tested on a water and a protein systems results were compared to those obtained using a langevin molly lm method 5 on the same systems the test results demonstrate the superiority of the new methods over lm in terms of stability accuracy and sampling efficiency this suggests that putting the mts approach in the framework of hybrid monte carlo and using the natural stochasticity offered by the generalized hybrid monte carlo lead to improving stability of mts and allow for achieving larger step sizes in the simulation of complex systems

an excursion into large rotations the present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in 1 it is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix t appertaining to a rotation about an arbitrary axis also analysed is the case of multiple rotations about fixed or follower axes particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations this theme is discussed in some detail for a number of cases semitangential rotations for which commutativity holds first proposed in 2 3 are also considered furthermore an elementary geometrical analysis of large rotations is also given finally we deduce in an appendix using a judicious reformulation of quarternions the compound pseudovector representing the combined effect of n rotations in the author s opinion the present approach appears preferable to a pure vectorial scheme and even more so to an indicial formulation and is computationally more convenient

gshmc an efficient method for molecular simulation the hybrid monte carlo hmc method is a popular and rigorous method for sampling from a canonical ensemble the hmc method is based on classical molecular dynamics simulations combined with a metropolis acceptance criterion and a momentum resampling step while the hmc method completely resamples the momentum after each monte carlo step the generalized hybrid monte carlo ghmc method can be implemented with a partial momentum refreshment step this property seems desirable for keeping some of the dynamic information throughout the sampling process similar to stochastic langevin and brownian dynamics simulations it is however ultimate to the success of the ghmc method that the rejection rate in the molecular dynamics part is kept at a minimum otherwise an undesirable zitterbewegung in the monte carlo samples is observed in this paper we describe a method to achieve very low rejection rates by using a modified energy which is preserved to high order along molecular dynamics trajectories the modified energy is based on backward error results for symplectic time stepping methods the proposed generalized shadow hybrid monte carlo gshmc method is applicable to nvt as well as npt ensemble simulations

kinematics of finite rotation shells with in plane twist parameter abstract the paper concerns shell equations derived from a 3d mixed formulation with explicit rotations the constitutive equation is postulated in terms of the relaxed biot stress and right stretch strain the shell kinematics is based on the reissner hypothesis and on a constant linear approximation of the rotation tensor with respect to the thickness coordinate the rotation around the shell director is parametrized by the drill and in plane twist parameters with the latter one associated with the linear approximation the in plane twist parameter is new and hence consequences of its presence for the shell rotation constrains and strain measures are examined and compared to the formulation with the drill rotation only in the presented formulation the drill rotation is retained explicitly and can be recovered from the drill rotation constraint while the in plane twist parameter is an internal parameter of the theory in this way the enhanced shell kinematics does not increase the number of degrees of freedom of a discrete model

a comparison of generalized hybrid monte carlo methods with and without momentum flip the generalized hybrid monte carlo ghmc method combines metropolis corrected constant energy simulations with a partial random refreshment step in the particle momenta the standard detailed balance condition requires that momenta are negated upon rejection of a molecular dynamics proposal step the implication is a trajectory reversal upon rejection which is undesirable when interpreting ghmc as thermostated molecular dynamics we show that a modified detailed balance condition can be used to implement ghmc without momentum flips the same modification can be applied to the generalized shadow hybrid monte carlo gshmc method numerical results indicate that ghmc gshmc implementations with momentum flip display a favorable behavior in terms of sampling efficiency i e the traditional ghmc gshmc implementations with momentum flip got the advantage of a higher acceptance rate and faster decorrelation of monte carlo samples the difference is more pronounced for ghmc we also numerically investigate the behavior of the ghmc method as a langevin type thermostat we find that the ghmc method without momentum flip interferes less with the underlying stochastic molecular dynamics in terms of autocorrelation functions and it to be preferred over the ghmc method with momentum flip the same finding applies to gshmc

an approximation to miscible fluid flows in porous media with point sources and sinks by an eulerian lagrangian localized adjoint method and mixed finite element methods we develop an eulerian lagrangian localized adjoint method ellam mixed finite element method mfem solution technique for accurate numerical simulation of coupled systems of partial differential equations pdes which describe complex fluid flow processes in porous media an ellam which was shown previously to outperform many widely used methods in the context of linear convection diffusion pdes is presented to solve the transport equation for concentration since accurate fluid velocities are crucial in numerical simulations an mfem is used to solve the pressure equation for the pressure and darcy velocity this minimizes the numerical difficulties occurring in standard methods for approximating velocities caused by differentiation of the pressure and then multiplication by rough coefficients the ellam mfem solution technique significantly reduces temporal errors symmetrizes the governing transport equation eliminates nonphysical oscillation and or excessive numerical dispersion in many simulators conserves mass and treats boundary conditions accurately numerical experiments show that the ellam mfem solution technique simulates miscible displacements of incompressible fluid flows in porous media accurately with fairly coarse spatial grids and very large time steps which are one or two orders of magnitude larger than the time steps used in many methods moreover the ellam mfem solution technique can treat large mobility ratios discontinuous permeabilities and porosities anisotropic dispersion in tensor form and point sources and sinks