Evaluation of integration methods for hybrid simulation of complex structural systems through collapse

Real-time hybrid testing (RTHT) involves virtual splitting of the structure into two parts: physical substructure that contains the key region of interest which is tested in a laboratory and numerical substructure that contains the remaining part of the structure in the form of a numerical model. This paper numerically assesses four step-by-step integration methods (Central difference method (CDM), Operator splitting method (OSM), Rosenbrock based method (RBM) and CR-integration method (CR)) which are widely used in RTHT. The methods have been assessed in terms of stability and accuracy for various realistic damping ratios of the physical substructure. The stability is assessed in terms of the spectral radii of the amplification matrix while the accuracy in terms of numerical damping and period distortion. In order to evaluate the performance of the methods, five carefully chosen examples have been studied - undamped SDOF, damped SDOF, instantaneous softening, instantaneous hardening and hysteretic system. The performance of the methods is measured in terms of a non-dimensional error index for displacement and velocity. Based on the error indices, it is observed that OSM and RBM are robust and performs fairly well in all the cases. CDM performed well for undamped SDOF system. CR method can be used for the system showing softening behaviour. The error indices indicate that accuracy of OSM is more than other method in case of hysteretic system. The accuracy of the results obtained through time integration methods for different damping ratios of the physical substructure is addressed in the present study. In the presence of a number of integration methods, it is preferable to have criteria for the selection of the time integration scheme. As such criteria are not available presently, this paper attempts to fill this gap by numerically assessing the four commonly used step-by-step methods.

In hybrid substructuring, a structural system is partitioned into a numerical substructure and a physical substructure. Typically, the physical substructure consists of a system component whose behavior is difficult to model while the numerical substructure consists of a computational model of the remainder of the system. Hybrid substructuring has previously been shown to be an effective method to quantify the effect of parametric uncertainties in the numerical substructure on the response of the system. This paper proposes and implements a methodology where the effect of parametric uncertainty can also be incorporated into the physical substructure. This idea is implemented in a series of small-scale Real-Time Hybrid Substructuring (RTHS) tests on a magneto-rheological fluid damper used to control a two degree-of-freedom mass-spring system. The physical current supplied to the damper is treated as a random variable. Using the RTHS test results, a metamodel of the system’s frequency domain behavior is developed using Principal Component Analysis and Kriging. This metamodel is then used to evaluate probabilistic system performance.

Proposed in this paper is an online hybrid test framework using a static and dynamic separated model scheme, where the equations of motion are solved by the operator-splitting (OS) algorithm, while the restoring force is obtained from a static analysis or quasi-static test of substructures. One of the keys of a successful substructure online hybrid test is the boundary coordination between the numerical and physical substructures. This is easily determined for the dynamic boundary whose displacement target is determined by the displacement predictor of the OS algorithm. However, the coordination is not straightforward for boundaries associated with static forces. This paper proposes an approximate method to determine the compatibility of the boundaries of static degrees of freedom, where the numerical substructure is analyzed first with the static boundary at the displacement of the previous step. The reaction force is then acquired from the analysis results as the target force of the corresponding degree of freedom of the physical substructure. The measured displacement is used as the target for the numerical substructure in the next step. In this procedure, the numerical substructure and the physical substructure alternatively move forward. Therefore, it is call staggered coordination. This is not a rigorous method to analyze boundary compatibility and equilibrium. However, it does make the substructure online hybrid test much easier and more feasible. Here, the staggered coordination scheme is examined by a six-story RC frame equipped with BRBs through both numerical simulation and physical testing. An energy-based index shows that the error introduced by this method can be ignored and that the proposed framework of online hybrid test works precisely without any malfunction.

Gain-margin based discrete-continuous method for the stability analysis of real-time hybrid simulation systems

An efficient method for computer simulations of liquid and amorphous systems is described. The method is based on replacing the natural trajectory of the system in phase space generated by the Newtonian equations of motion by a Markov chain. Convergence to the Gibbs distribution is proved. The relation to the other simulation methods is discussed. The method is applied to simulation of the structure of amorphous iron. As an example of employing the method to study thermodynamic properties, the quenching of the system is investigated.

The random batch Ewald (RBE) is an efficient and accurate method for molecular dynamics (MD) simulations of physical systems at the nano/microscale. The method shows great potential to solve the computational bottleneck of long-range interactions, motivating a necessity to accelerate short-range components of the nonbonded interactions for a further speedup of MD simulations. In this work, we present an improved RBE method for the nonbonding interactions by introducing the random batch idea to constructing neighbor lists for the treatment of both the short-range part of the Ewald splitting and the Lennard-Jones potential. The efficiency that the novel neighbor list algorithm owes to the stochastic minibatch strategy can significantly reduce the total number of neighbors. We obtain the error estimate and convergence by theoretical analysis and implement the improved RBE method in the LAMMPS package. Benchmark simulations are performed to demonstrate the accuracy and stability of the algorithm. Numerical tests on computer performance by conducting large-scaled MD simulations for systems including up to 0.1 billion water molecules are run on massive clusters with up to 50000 CPU cores, demonstrating the attractive features such as the high parallel scalability and memory-saving of the method in comparison to the existing methods.

Understanding the limiting capabilities of classical methods in simulating complex quantum systems is of paramount importance for quantum technologies. Although many advanced approaches have been proposed and recently used to challenge quantum advantage experiments, novel efficient methods for the approximate simulation of complex quantum systems are still in high demand. Here, we propose a scalable technique for approximate simulations of intermediate-size quantum circuits on the basis of the multi-scale entanglement renormalization ansatz (MERA) and Riemannian optimization. The MERA is a tensor network, whose geometry together with orthogonality constraints imposed on its tensors allow approximating many-body quantum states lying beyond the area-law scaling of the entanglement entropy. We benchmark the proposed technique for brick-wall quantum circuits of up to 243 qubits with various depths up to 20 layers. Our approach paves a way to exploring efficient simulation techniques for quantum many-body systems.

A new method for efficient dynamics simulations of flexible systems with unilateral contacts is presented. The procedure is based on the Craig-Bampton model-reduction technique together with a contact formulation using single- and set-valued force laws. For the computation of system response, an event-driven integration scheme is proposed, which allows transitions among different contact states. An updating algorithm for derivation of the reduced-system matrices is developed as well as the formulation of new initial conditions. The applicability of the developed method is demonstrated on a simple structure with varying contact conditions.

Three multiscale computational methodologies for biomolecular systems are described: the force-matching method for developing coarse-grained models directly from atomistic simulations; the quasi-particle approach of simulating field theory representations at the mesoscopic scale; and the multiscale-coupling method for direct information transfer between mesoscopic and atomistic scales on the fly. The statistical mechanical background for each of the methods is described in a comprehensive manner in order to highlight their theoretical foundations. Examples of various applications of these methods to model different biophysical processes are given. Combining with atomistic-level MD simulations, these three methods compose a powerful tool for bridging and spanning the multiple spatial and temporal domains that are present in many biological assemblies. Future directions of the methodology developments are also discussed.

Stochastic modelling is critical for studying many biochemical processes in a cell, in particular when some reacting species have low population numbers. For many such cellular processes the spatial distribution of the molecular species plays a key role. The evolution of spatially heterogeneous biochemical systems with some species in low amounts is accurately described by the mesoscopic model of the Reaction-Diffusion Master Equation. The Inhomogeneous Stochastic Simulation Algorithm provides an exact strategy to numerically solve this model, but it is computationally very expensive on realistic applications. We propose a novel adaptive time-stepping scheme for the tau-leaping method for approximating the solution of the Reaction-Diffusion Master Equation. This technique combines effective strategies for variable time-stepping with path preservation to reduce the computational cost, while maintaining the desired accuracy. The numerical tests on various examples arising in applications show the improved efficiency achieved by the new adaptive method.

We present a recently developed multigrid calculation scheme for coupled magnetomechanical systems. The scheme allows the efficient calculation of three-dimensional (3-D) dynamic rigid motions as well as deformations of nonmagnetic and ferromagnetic materials in a magnetic field. Thereby, the 3-D mechanical problem is discretized with nodal elements and the 3-D magnetic problem with edge elements. Both finite element meshes may be chosen independently and thus can be separately adapted to the physical requirements of the mechanical and magnetic fields. Fast multigrid solution techniques can solve large scaled 3-D problems within very short CPU times. Three examples demonstrate the efficiency of the developed scheme.

Computational and Systems Biology are recently experiencing a rapid development. Mathematical modeling is a key tool in analyzing critical biological processes in cells and living organisms. In particular, stochastic models are essential to accurately describe the biochemical processes in the cell. However, stochastic models are computationally much more challenging than the traditional deterministic models. Also, the sheer scale of biological processes makes efficiency of the simulation a key issue. In this thesis we study the numerical solution of a continuous stochastic model of biochemical systems, the Chemical Langevin Equation. We propose an adaptive step-size method for the Euler-Maruyama scheme applied to small noise problems. The adaptive technique is p-mean convergent and computes simultaneously all the trajectories, using the same time-step on all trajectories. Our adaptive algorithm is tested on several examples arising in applications and shown to perform much better than the fixed step-size schemes.

Dynamically substructured system testing method divides an original system into several substructures. In the numerical substructure, linear components are simulated via real-time computation. In the physical substructure, a transfer system, which includes actuators and sensors, is installed to interface the numerical and physical parts. During the test, the unwanted disturbances and noise from the actuator inevitably cause synchronization errors between the outputs of numerical and physical substructures at the interface, and consequently result in unsuccessful tests. Therefore, this study proposes advanced control using the feedforward state-space linear substructuring controller plus the mixed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> /H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> feedback controller to ensure optimal and robust synchronization. The synchronization problem is transformed to tracking design according to a numerical-substructure-based framework and is solved based on Riccati-like equations and linear matrix inequality. A multivariable mass-spring-damper substructured system is developed to verify the proposed control strategy via numerical studies.

The hybrid simulation (HS) testing method combines physical test and numerical simulation, and provides a viable alternative to evaluate the structural seismic performance. Most studies focused on the accuracy, stability and reliability of the HS method in the small-scale tests. It is a challenge to evaluate the seismic performance of a twelve-story pre-cast reinforced concrete shear-wall structure using this HS method which takes the full-scale bottom three-story structural model as the physical substructure and the elastic non-linear model as the numerical substructure. This paper employs an equivalent force control (EFC) method with implicit integration algorithm to deal with the numerical integration of the equation of motion (EOM) and the control of the loading device. Because of the arrangement of the test model, an elastic non-linear numerical model is used to simulate the numerical substructure. And non-subdivision strategy for the displacement inflection point of numerical substructure is used to easily realize the simulation of the numerical substructure and thus reduce the measured error. The parameters of the EFC method are calculated basing on analytical and numerical studies and used to the actual full-scale HS test. Finally, the accuracy and feasibility of the EFC-based HS method is verified experimentally through the substructure HS tests of the pre-cast reinforced concrete shear-wall structure model. And the testing results of the descending stage can be conveniently obtained from the EFC-based HS method.

Real-time hybrid simulation (RTHS) is a promising cyber-physical technique for the experimental evaluation of civil infrastructure systems subject to extreme dynamic loading. It allows for simulation of highly complicated civil infrastructure systems in a cost effective manner by partitioning them into numerical and physical substructures. And the coupling at the interface is achieved by enforcing equilibrium and compatibility at real-time. In RTHS, development of some systematic and random errors in the physical substructure are inevitable and can have substantial impacts on the quality of the simulation results. It has been shown that the reliability of RTHS results is highly dependent upon how successfully these sources of error in the physical substructure are mitigated. Some significant sources of error in RTHS are transfer system dynamics, e.g. dynamics of hydraulic actuator, and time delay, e.g. communication and computation delays. In this paper, we will show how a multi-degree of freedom (MDOF) RTHS system can be modeled with delay differential equations (DDE). Afterwards, a 4-story RTHS will be used as an illustrative numerical example to delineate how this DDE model can capture some important features of RTHS. Finally, the effects of transfer system dynamics on the quality of RTHS results will be shown for a 2-story structure and then the DDE model will be verified.