Large-scale Structural Dynamics Research Articles

The application of single-molecule optical tweezers to study disease-related structural dynamics in RNA.

RNA, a dynamic and flexible molecule with intricate three-dimensional structures, has myriad functions in disease development. Traditional methods, such as X-ray crystallography and nuclear magnetic resonance, face limitations in capturing real-time, single-molecule dynamics crucial for understanding RNA function. This review explores the transformative potential of single-molecule force spectroscopy using optical tweezers, showcasing its capability to directly probe time-dependent structural rearrangements of individual RNA molecules. Optical tweezers offer versatility in exploring diverse conditions, with the potential to provide insights into how environmental changes, ligands and RNA-binding proteins impact RNA behaviour. By enabling real-time observations of large-scale structural dynamics, optical tweezers emerge as an invaluable tool for advancing our comprehension of RNA structure and function. Here, we showcase their application in elucidating the dynamics of RNA elements in virology, such as the pseudoknot governing ribosomal frameshifting in SARS-CoV-2.

A new time integration method based on state formulations for dynamic analysis of nonviscously damped systems

The mode superposition method (MSM), which is used for dynamic response calculations of nonviscously damped systems, requires state–space eigensolutions, which incur a high computational cost. This study proposed an alternative symmetric formulation-based time integration method for analyzing structural dynamic systems. An anelastic displacement field (ADF) model was introduced to capture damping mechanisms in nonviscous systems. The differential equation describing the ADF model was transformed into an augmented coupled system via the introduction of dissipative variables. Two symmetric state formulations based on coupled augmented systems were derived for nonviscous systems. Consequently, alternative asymmetric state formulations were derived for nonviscous systems involving the ADF model. Thereafter, a new method was developed based on these state–space formulations by assuming linear approximations. A numerical implementation of the new method for dynamic systems was demonstrated. Three benchmark examples of discrete and continuous systems were considered to evaluate the asymmetric formulation and the computational efficiency, accuracy, stability, and time-step-size sensitivity of the new method. Further, the response of the new method was validated using MSM and precise time integration (PTI) methods. An asymmetric state formulation was also validated against MSM and direct frequency response methods. The results revealed that the new method accurately matched the MSM and PTI methods at various time steps. The results also showed that the new method was more efficient than the MSM and PTI methods, rendering it suitable for solving large-scale structural dynamics problems.

Hydrodynamic coefficients of mussel dropper lines derived from large-scale experiments and structural dynamics

The expansion of marine aquaculture production is driven by a high market demand for marine proteins and a stagnation of wild catch of fish. Bivalve farming, i.e., the cultivation of oysters, mussels and scallops, is an important part of the ongoing market dynamics and production expansion. As marine spatial planning is considering various use purposes, available space for near-shore aquaculture is already becoming scarce; this has fueled research and development initiatives to enable production installations further offshore. The highly energetic conditions at more exposed offshore marine sites lead to increased loads on aquaculture systems and their components and it is still not sufficiently understood how the load transfer from oceanic environmental conditions onto shellfish-encrusted surfaces attached to elastic ropes may be appropriately quantified. This study data gathered large-scale data sets in a wave tank facility, which are used to validate a novel, numerical model, building on the dynamics of rope structures which allows for the determination of the hydrodynamic loads transferred to the dropper lines. The forces and hydrodynamic parameters are measured and numerically analyzed. Based on the results, drag and inertia coefficients are determined. A drag coefficient of CD=1.1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_\ extrm{D} = 1.1$$\\end{document} and an inertia coefficient of CM=1.7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_\ extrm{M} = 1.7$$\\end{document} are recommended to model shellfish-encrusted dropper lines exposed to oscillatory flows with KC =\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$=$$\\end{document} 40–90. The numerical model for the determination of wave-induced forces on mussel dropper lines is developed and validated using the experimental data. It employs a modified Morison equation, which takes into account the displacement of the mussel dropper line. The influence of varying aquaculture-related parameters is discussed by applying the numerical model. Based on the gathered insights, recommendations can be given from an engineering point of view concerning the optimal placement of mussel aquaculture within the water column.

Quantifying the large-scale chromosome structural dynamics during the mitosis-to-G1 phase transition of cell cycle.

Cell cycle is known to be regulated by the underlying gene network. Chromosomes, which serve as the scaffold for gene expressions, undergo significant structural reorganizations during mitosis. Understanding the mechanism of the cell cycle from the chromosome structural perspective remains a grand challenge. In this study, we applied an integrated theoretical approach to investigate large-scale chromosome structural dynamics during the mitosis-to-G1 phase transition. We observed that the chromosome structural expansion and adaptation of the structural asphericity do not occur synchronously and attributed this behaviour to the unique unloading sequence of the two types of condensins. Furthermore, we observed that the coherent motions between the chromosomal loci are primarily enhanced within the topologically associating domains (TADs) as cells progress to the G1 phase, suggesting that TADs can be considered as both structural and dynamical units for organizing the three-dimensional chromosome. Our analysis also reveals that the quantified pathways of chromosome structural reorganization during the mitosis-to-G1 phase transition exhibit high stochasticity at the single-cell level and show nonlinear behaviours in changing TADs and contacts formed at the long-range regions. Our findings offer valuable insights into large-scale chromosome structural dynamics after mitosis.

Intrinsically Disordered Regions Function as a Cervical Collar to Remotely Regulate the Nodding Dynamics of SARS-CoV-2 Prefusion Spike Heads.

The SARS-CoV-2 prefusion spike heads (receptor binding domains, RBDs) frequently nod down and up to interact with host cell receptors. As the spike protein is a trimeric unit of significant size, to understand its large-scale structural dynamics associated with the nodding mechanism and the mutational impact on the same, we develop a topological symmetry-information-loaded coarse-grained structure-based model of a spike trimer using recent cryo-EM structural data. Our study reveals the control of two distant intrinsically disordered regions (IDRs), namely, 630 and FPPR loops, over the nodding dynamics of spike heads. We find that the order-disorder transition of IDRs becomes more evident in the variants of concern (VOCs) that are associated with the characteristic mutation, D614G, in the proximity of these IDRs. In some VOCs, the two other mutations A570D and S982A also show an integral effect. The driver mutation D614G instigates a salt-bridge disruption, altering the order-disorder dynamics of both 630 and FPPR loops and their interaction with the C-terminal domains (CTD1/CTD2). This altered connectivity in these mutants allows the two IDRs to act collectively as a "cervical collar" for the RBD, supporting various spike head postures, consistent with cryo-EM results available for specific cases. The IDRs' control over the spike structure and dynamics presents an exciting opportunity where they can be targeted as remote operational switches to artificially maneuver the nod for effective therapeutic interventions.

Geomorphometric characterization and sediment connectivity of the middle Brahmaputra River basin

Knowledge of geomorphic characteristics and sediment connectivity is critical in understanding basin-scale surface processes that include steep mountainous and flat lowlands, particularly if the mid-basin and downstream areas are densely populated and disaster-prone. However, the lack of fluvial and sediment connectivity studies in downstream portions of large rivers of South Asia (particularly Bangladesh), where upper basin morphodynamics, hydrology, and sediment flux highly influence the repeated sediment disasters, limits our understanding of the large-scale fluvial geomorphic processes and basin-scale sustainable river management. Therefore, this article attempts to characterize the morphometric and topographic features along with the spatial sediment connectivity pattern of the middle Brahmaputra River basin (Teesta, Torsa, and Manas basins) covering ~75,000 km2 that influences the hydro-geomorphic response of the deltaic landscape of lowland Bangladesh. This work considers a set of morphometric and topographic parameters including the stream network, longitudinal profiles, stream power index (SPI), and topographic wetness index (TWI) for geomorphometric characterization. The linkage between the sediment sources to downstream areas also has been evaluated with the sediment connectivity (IC) index to understand better the sediment dynamics of the middle Brahmaputra River basin, draining towards lowland Bangladesh from the basin's upstream countries of India and Bhutan. The result of this study demonstrates a highly potential hydro-geomorphic response of the downstream areas attributed to steep topography, steep channel longitudinal profiles, high rainfall, and high sediment connectivity in upstream regions. However, the low topography, presence of anthropogenic stressors, almost flat longitudinal channel profiles with limited change in elevation, and lowered sediment connectivity potential in the mid-to-downstream areas depicts sensitivity to the depositional processes therein, impacting the basin-scale geomorphic connectivity from the upstream to the downstream region. These results will be the basic information for exploring the large-scale hydro-geomorphic response and structural sediment dynamics to understand the complex geomorphological processes in the South Asia region, where the fluvial sedimentation process is not fully understood with limited accessibility for field-based investigations.

Time-varying elastic wave mode conversion in vibrating elastic beams with subwavelength nonlinearity

Introduction of time-varying properties in a background medium can enable unconventional wave behavior. Furthermore, it is well-known that subjecting nonlinear materials to time-varying stress can be used for material characterization [Appl. Phys. Lett. 94, 011905 (2009)]. We propose a technique called Dynamic Asymmetric Transmission Measurement (DATM) that combines structural dynamic (SD) and ultrasonic (US) testing modalities in an elastic waveguide to detect and characterize changes in the global stress state of a structure that result in local time-varying stress conditions due to local nonlinearity. Asymmetric geometric features in beams and plates, paired with time-varying stress conditions, result in asymmetric mode conversion of guided US waves that depend on large-scale structural dynamics. The DATM technique is explored through finite element modeling, semi-analytical methods, and experiments and discussed in a structural health monitoring context. We also discuss the use of dynamic structures with engineered defects to introduce time-varying stress conditions for the purpose of manipulating elastic waves. [Sandia National Laboratories is a multimission laboratory managed and operated by the National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.]

A technique for measurement of ultrasonic waves propagating in time-varying media

Laser Doppler Vibrometry (LDV) is often used in nondestructive testing to make high-fidelity, non-contact structural dynamic measurements. However, synchronized measurement on slow, structural dynamics (SD) and fast, ultrasonic (US) time scales can yield additional useful information about the overall condition of elastic structures. This work presents a technique, Dynamic Asymmetric Transmission Measurement (DATM), that enables time-aligned measurements of large-scale structural dynamics via LDV and small-scale ultrasonic signals to provide information at a particular global dynamic state of an elastic structure. This approach is applicable to structural health monitoring and generalizable to the measurement of guided ultrasonic waves propagating in structural components subjected to time-varying stress states. We demonstrate the utility of DATM by detecting and characterizing time-varying stress conditions caused by local nonlinearity due to surface-breaking cracks resulting in time-varying asymmetric transmission of US signals in beam and plate structures. The test methodology to synchronize LDV and US data is presented together with example measurements. [Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.]

Particle Filter Method to Integrate High-Speed Atomic Force Microscopy Measurements with Biomolecular Simulations.

High-speed atomic force microscopy (HS-AFM) can be used to observe the structural dynamics of biomolecules at the single-molecule level in real time under near-physiological conditions; however, its spatiotemporal resolution is limited. Complementarily, molecular dynamics (MD) simulations have higher spatiotemporal resolutions, albeit with some artifacts. Here, to integrate HS-AFM data and coarse-grained molecular dynamics (CG-MD) simulations, we develop a particle filter method that implements a sequential Bayesian data assimilation approach. We test the method in a twin experiment. First, we generate a reference HS-AFM movie from the CG-MD trajectory of a test molecule, a nucleosome; this serves as the "experimental measurement". Then, we perform a particle filter simulation with 512 particles, which captures the large-scale nucleosome structural dynamics compatible with the AFM movie. Comparing particle filter simulations with 8-8192 particles, we find that using greater numbers of particles consistently increases the likelihood of the whole AFM movie. By comparing the likelihoods for different ionic concentrations and time scale mappings, we find that the "true" concentration and time scale mapping can be inferred as the largest likelihood of the whole AFM movie but not that of each AFM image. The particle filter method provides a general approach for integrating HS-AFM data with MD simulations.

State-space approach for transverse vibration of double-beam systems

As a basic mechanical model of modern beam-type structures, double-beam systems have intensively been studied in recent decades including the structural vibration and energy dissipation with consideration of damping interlayer inside double beams. However, the precision and calculation efficiency of those methods have limitations for modeling and simulation such as large-scale structural dynamics and vibration control. This paper presents a novel state-space approach aiming to improve the calculation accuracy and efficiency. A generalized double-beam system is defined with the modified governing equations of motion, followed by a proposed mode-shape constant to introduce the state space in modeling framework. Based on new state-space equations, transverse vibration of the double-beam system can be analyzed adequately. Numerical examples demonstrate that the proposed method is accurate and fast, making it qualified for large-scale modeling and simulation for various engineering structures.

Flexible Fitting of Biomolecular Structures to Atomic Force Microscopy Images via Biased Molecular Simulations.

High-speed (HS) atomic force microscopy (AFM) is a prominent imaging technology that observes large-scale structural dynamics of biomolecules near the physiological condition, but the AFM data are limited to the surface shape of specimens. Rigid-body fitting methods were developed to obtain molecular structures that fit to an AFM image, without accounting for conformational changes. Here, we developed a method to fit flexibly a three-dimensional (3D) biomolecular structure into an AFM image. First, we describe a method to produce a pseudo-AFM image from a given 3D structure in a differentiable form. Then, using a correlation function between the experimental AFM image and the computational pseudo-AFM image, we developed a flexible fitting molecular dynamics (MD) simulation method by which we obtain protein structures that well fit to the given AFM image. We first test it with a twin experiment; using an AFM image produced from a protein structure different from its native conformation as a reference, we performed the flexible fitting MD simulations to sample conformations that fit well the reference AFM image, and the method was confirmed to work well. Then, parameter dependence in the protocol was discussed. Finally, we applied the method to a real experimental HS-AFM image for a flagellar protein FlhA, demonstrating its applicability. We also test the rigid-body fitting of a molecular structure to an AFM image. Our method will be a general tool for dynamic structure modeling based on HS-AFM images and is publicly available through the CafeMol software.

Structural Dynamics Couple Substrate Recognition with Allosteric Domain Communication in Nonribosomal Peptide Synthetases

Timely control of protein interactions is central to modulate biological activity such as gene activation, cell signaling or metabolism. Here, events such as post‐translational modifications and substrate or protein binding affect proteins so as to govern their interactions with partner proteins and repress or promote activity. In spite of progress in identifying protagonists of molecular communication and predicting the outcome of molecular events, it has proven more challenging to determine how molecular responses within proteins translate into responses between proteins, particularly when remote sites are involved within proteins. Such challenges are inherent to understanding microbial, multi‐domain enzymatic systems called nonribosomal peptide synthetases (NRPSs). NRPSs employ contiguous repeats of domains to covalently tether substrates onto carrier proteins (CP) and assemble them into complex products through intervening condensation or cyclization domains. The pharmaceutical properties of NRPS products (e.g. antibiotics) have spurred attempts at engineering NRPSs, but dynamics between domains have hampered progress, as it remains unclear whether and how the many catalytic steps required for synthesis relate with transient domain communication. Using NMR, we demonstrate that large scale structural dynamics within a 52 kDa cyclization domain sense substrates tethered to partner CPs to not only promote binding between both partner domains but also to remodel both the active site and a remote binding site for downstream partners, thus revealing a path for allosteric communication involving three partner domains. We provide a 3D structural description of the enzyme dynamics and demonstrate that mutations that impede catalysis injure the same dynamic allosteric network, suggesting that catalysis efficiency may be streamlined with productive domain communication and substrate recognition through dynamic responses. Our results reveal and describe at the molecular level a substrate driven allosteric response in NRPSs, and they exemplify how structural dynamics within proteins can couple substrate recognition with active site remodeling and promotion of both local and remote protein communication.Support or Funding InformationNIH‐R01GM104257This abstract is from the Experimental Biology 2019 Meeting. There is no full text article associated with this abstract published in The FASEB Journal.

An explicit asynchronous step parallel computing method for finite element analysis on multi-core clusters

The finite element analysis of complex structure often requires a refine mesh in some local domain. To reduce the computation time, an explicit asynchronous step parallel computing method is proposed. The domain decomposition with overlapping node method is used to divide the model into different subdomains. The multiple overlapping nodes between different subdomains constitute the coupling region. Each subdomain selects the time step based on the mesh characteristics. The subcycling method is adopted to tackle the matching of asynchronous step boundary. The subdomain model, boundary information and calculation results are stored in parallel files which mean the overall finite element analysis process is implemented in parallel. The validity and efficiency of the proposed method are verified through three simulation cases which conducted on Tianhe 2 multi-core supercomputers. The results of simulation cases show that the proposed method has a higher accuracy than classic subcycling method under the same time step. The total speedup of the algorithm relates to the step ratios between subdomains, the number of subdomain and the load balance. This approach offers an efficient way to solve large-scale and super-scale structural dynamics analysis with local refine mesh.

Fast algorithm for structural dynamics problems using differential quadrature method and $${\varvec{V}}$$ V -transformation

The differential quadrature method (DQM) has been widely used for structural dynamics problems computation. Traditional DQM usually adopts simultaneous discretization in multi-grid points, and then all variables are solved simultaneously. Obviously, this will result in the curse of dimensionality problem for large-scale systems. In this paper, on the basis of the multi-stage high-order time domain DQM, a fast numerical calculation method for large-scale structural dynamics problems based on $${\varvec{V}}$$ -transformation is proposed. Using the $${\varvec{V}}$$ -transformation possessed by the weighting coefficient matrix of DQM, the whole Jacobian matrix equations involved in the traditional approach of DQM can be decoupled into blocks; thus, the multi-stage block recursive method is derived. Numerical experiments show that: even using 2s times step size of the Newmark method, the computational accuracy of DQM is about 2–3 orders of magnitude higher than that of the Newmark method. Furthermore, three different scale systems are used for time test and the results show that the multi-stage block recursive method can obtain high speedup ratio, which can significantly improve the computational efficiency of large-scale structural dynamics problems.

Free Energy Landscapes for a Kainate Receptor Ligand-Binding Domain

The kainate family of ionotropic glutamate receptors (GluK) mediate neurotransmission in pre- and post-synaptic neurons. Previously computed free energy landscapes for iGluR ligand-binding domains (LBD) revealed subtype-dependent conformational differences between NMDA and AMPA iGluR families. Here we report free energy landscapes for the GluK2 ligand-binding domain (LBD) in apo and glutamate-bound states. The GluK2 free energy landscapes reveal both apo and glutamate-bound LBDs preferentially access the closed-cleft conformation and suggest a conformational selection mechanism of binding. These results are consistent with the free energy landscapes for NMDA receptors but not AMPA receptors. Additionally, the open-cleft conformation is more accessible to the GluK2 apo state than the glutamate-bound state, consistent with previous results for both NMDA and AMPA receptors. Finally, large-scale structural dynamics are characterized by principle component analysis.

A fast precise integration method for structural dynamics problems

A fast precise integration method (FPIM) is proposed for solving structural dynamics problems. It is based on the original precise integration method (PIM) that utilizes the sparse nature of the system matrices and especially the physical features found in structural dynamics problems. A physical interpretation of the matrix exponential is given, which leads to an efficient algorithm for both its evaluation and subsequently the solution of large-scale structural dynamics problems. The proposed algorithm is accurate, efficient and requires less computer storage than previous techniques.

An iterative domain decomposition method for the solution of a class of indefinite problems in computational structural dynamics

The FETI-DP domain decomposition method (DDM) is extended to address the iterative solution of a class of indefinite problems of the form ( A − σ M ) x = b , where A and M are two real symmetric positive semi-definite matrices arising from the finite element discretization of second-order elastodynamic problems, and σ is a positive number. A key component of this extension is a new coarse problem based on the free-space solutions of Navier's homogeneous displacement equations of motion. These solutions are waves, and therefore the resulting DDM is reminiscent of the FETI-H method. For this reason, it is named here the FETI-DPH method. For a given σ, this method is numerically shown to be scalable with respect to all of the problem size, subdomain size, and number of subdomains. Its intrinsic CPU performance is illustrated for various ranges of σ with the solution on an Origin 3800 parallel processor of several large-scale structural dynamics problems.

Explicit unconditionally stable approaches for built-up shell structural configurations

Built-up structures, especially those involving shell-type components, are encountered in many areas of engineering. Since full three-dimensional modeling may be cost prohibitive, shell-type elements have played an important role in dynamic simulations ; however, the nonlinear dynamic analysis is still relatively expensive because of the enormous computations involved. Most often, implicit approaches such as the Newmark β = 0.25 are commonly employed. With the motivation of further enhancing the accuracy and efficiency of analysis of large practical structural problems, we describe explicit, unconditionally stable approaches newly developed by the authors to analyze the dynamics of linear/nonlinear shell structures and subsequently show the applicability to large-scale practical structural dynamics problems. The explicit nature of the formulations and the unconditionally stable algorithmic stability and excellent algorithmic attributes in conjunction with efficient numerical computational features indeed lend themselves well for the analysis of a wide class of complex shell-type structural configurations. The computational and implementation aspects and the numerical evaluation of the so-called VIP (virtual-pulse) time integral methodology that inherits these attributes for general shell-type structural dynamics problems are presented here. Comparisons are also drawn between the VIP methodology and the Newmark family of methods on the aspects of the accuracy and computing time. The numerical results given, which were performed on the Cray supercomputer, show the applicability of the VIP methodology for practical problems, and the computations demonstrate the significant reduction in computing time compared with the most widely advocated Newmark family of methods for given accuracy conditions.

A method of deriving parallel algorithms for direct integration in structural dynamics

A method of deriving parallel algorithms from well-established conventional direct integration algorithms such as the Wilson θ method, Newmark β method etc. is presented. The new algorithms are able to effectively exploit the power of parallel computers for the dynamic analysis of large-scale structures, and may be more efficient than the conventional methods, even on uniprocessors. It appears that the parallel algorithms can be made to maintain the property of unconditional stability displayed by the conventional methods. The parallel algorithms are derived by splitting the stiffness, damping and mass matrices such that the dynamic equations of motion can be cast in block diagonal form. The splitting of the matrices corresponds to physical subdomains of the structure. Each set of block diagonal equations can be solved independently of the others and therefore the computations can be performed in parallel. However, a predictor-corrector approach requiring iterations within a time step must be used for acceptable accuracy. The method described in this paper is derived from the Wilson θ method and is called the “parallel Wilson θ method.” A simple numerical example is used to illustrate the accuracy of the parallel method, which is comparable to the conventional method when a sufficient number of iterations is used. Preliminary computations were performed on a BBN GP-1000 distributed-memory parallel computer to assess the performance of the parallel algorithm. The proposed method is especially suited for large-scale non-linear structural dynamics problems.

Parallel-vector design sensitivity analysis in structural dynamics

This paper presents a parallel-vector algorithm for sensitivity calculations in linear structural dynamics. Theoretical aspects of an alternative formulation for design sensitivity analysis in linear structural dynamics have recently been presented by the authors. The proposed alternative formulation works efficiently with the reduced system of dynamic equations, since it eliminates the need for expensive and complicated based-vector derivatives, which are required in the conventional reduced system formulation. The relationship between the alternative formulation and the conventional reduced system formulation has been established, and it has been proven analytically that the two approaches are identical when all the mode shapes are included. This paper validates the proposed alternative algorithm through numerical experiments, where only a small number of mode shapes are used. In addition, a modified mode acceleration method is presented, thus not only the displacements but also the velocities and accelerations are shown to be improved. An efficient parallel-vector algorithm for design sensitivity analysis in large-scale structural dynamics is developed, which is then implemented in a parallel FORTRAN (Force) code, PVDSASD. The efficiency of the algorithm is illustrated by analyzing practical structural systems, using the CRAY-2 and CRAY-YMP supercomputers, and making comparison with the sequential version of the algorithm.