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Hinged booms are widely used in astrophysics missions; however, the trajectory and deployment velocity are difficult to control because they are usually driven by springs, which limits their application in narrow spaces. Thus, a novel hinged boom is highly required to achieve motion controllability. Through an equivalent substitution between the cable drive loop and the binary link in topology, a type synthesis method for the cable-driven single-degree-of-freedom chain is proposed based on the single-open-chain (SOC) adding method. According to the configuration design, a novel cable-driven hinged boom is proposed, aiming to achieve boom synchronism. Then, to preload easily, a method that preload is applied and measured at the cable ends is adopted and the relationship between the initial preload and the target preload is deduced. By analyzing the distribution of cable tension, a new stiffness model is proposed thus a stiffness equation is obtained. Finally, the dynamic simulation analysis and zero-gravity deployment experiment of the hinged boom is carried out to verify its reliability. This research provides a new way for the type synthesis of cable-driven single-degree-of-freedom chain and a new model for analyzing cable-driven stiffness. Moreover, the novel cable-driven hinged boom obtained in this study can be well-applied in the field of aerospace.

A new numerical method for solving systems of ordinary differential equations (ODEs) by reducing them to Shannon’s equations is considered. To transform the differential equations given in the normal Cauchy form to Shannon’s equations, it is sufficient to perform a simple change of variables. Nonlinear ODE systems are linearized. Piecewise linear approximation of the right-hand sides of the Shannon equations does not require the calculation of the Jacobi matrix and ensures the accurate solution of differential equations, including stiff differential equations.

The recently developed physics-informed neural network (PINN) has achieved success in many science and engineering disciplines by encoding physics laws into the loss functions of the neural network such that the network not only conforms to the measurements and initial and boundary conditions but also satisfies the governing equations. This work first investigates the performance of the PINN in solving stiff chemical kinetic problems with governing equations of stiff ordinary differential equations (ODEs). The results elucidate the challenges of utilizing the PINN in stiff ODE systems. Consequently, we employ quasi-steady-state assumption (QSSA) to reduce the stiffness of the ODE systems, and the PINN then can be successfully applied to the converted non-/mild-stiff systems. Therefore, the results suggest that stiffness could be the major reason for the failure of the regular PINN in the studied stiff chemical kinetic systems. The developed stiff-PINN approach that utilizes QSSA to enable the PINN to solve stiff chemical kinetics shall open the possibility of applying the PINN to various reaction-diffusion systems involving stiff dynamics.

Simulating a physical system in real-time is widely used in equipment design, test, and validation. Though an implicit multistep numerical method excels at solving physical models that are usually composed of stiff ordinary differential equations, it is not suitable for real-time simulation because of state discontinuity and massive iterations for root finding. Thus, a method based on the backward differential formula is presented. It divides the main fixed step of real-time simulation into limited minor steps according to computing cost and accuracy demand. By analyzing and testing its capability, this method shows advantage and efficiency in real-time simulation, especially when the system contains stiff equations. A simulation application will have more flexibility while using this method.

A multi-scale method based on a combination of the boundary element method (BEM) and peridynamics (PD) was developed to model crack propagation problems in two-dimensional (2D) elastic bodies. The special feature of this method is that it can take full advantage of both the BEM and PD to achieve a higher level of computational efficiency. Based on the scale of the structure and the crack location, the considered domain can be divided into non-cracked and cracked domains. The BEM is employed in the non-cracked domain, while the PD is applied in the cracked domain. This can reduce the dimension by one in the non-cracked domain for improving the modeling efficiency. A stiffness equation of the bond-based PD is established by using Taylor’s series expansion for the bond stretch and applied to simulate the cracked domain. The PD approach can automatically model the initiation and propagation of a crack. A coupling model using shared nodes is constructed by introducing the BEM nodes on the interface at the same location as the PD material points. With the continuity of displacements and equilibrium of tractions at the interface, a combined system of equations is obtained by merging the stiffness and force matrix from each domain. For test problems, the deformation and crack propagation in 2D elastic bodies subjected to quasi-static loads were analyzed. The numerical results clearly demonstrate the accuracy and efficiency of the proposed method for crack problems based on coupling the BEM and PD.

Variable stepsize, variable order (VSVO) methods have limited impact in timestepping methods in complex applications due to their computational complexity and the difficulty in implementing them in...

This paper aims at constructing the first variable-stepsize doubly quasi-consistent singly diagonally implicit two-step peer pairs equipped with local error control for accurate and efficient numerical integration of stiff ordinary differential equations (ODEs). The main novelty of the numerical integration tools under consideration is that these accommodate the property of double quasi-consistency to variable meshes. This property means that the principal terms of the local and global errors coincide. It improves greatly the accuracy of conventional local error control facility and even translates it into a global one when the influence of the remaining terms in the mentioned error expansions is negligible. Our study proves theoretical conditions, which ensure the double quasi-consistency in the family of variable-stepsize singly diagonally implicit two-step peer formulas, and presents a feasible procedure for constructing such numerical schemes in practice. In particular, two embedded doubly quasi-consistent pairs of convergence order 3 with local error control are built in this way. Our numerical examples confirm their high efficiency for treating stiff ODEs, including large-scale systems obtained by semidiscretization of partial differential equations (PDEs). Moreover, the numerical integration tools designed here are shown to outperform the well-optimized Matlab code ODE23s with local error control, which is also a stiff ODE solver of order 3. An exhaustive comparison to the other built-in Matlab code ODE15s with local error control, which is considered to be a benchmark means for solving stiff ODEs by many practitioners, is also fulfilled and commented.

Numerical modeling of nonequilibrium state-to-state carbon dioxide kinetics is a challenging time-consuming computational task that involves solving a huge system of stiff differential equations and requires optimized methods to solve it. In the present study, we propose and analyse optimizations for the Extended Backward Differential Formula (EBDF) scheme. Using adaptive timesteps instead of fixed ones reduces the number of steps in the algorithm many thousands of times, although with an increase in step complexity. The use of parallel computations to calculate relaxation terms allows one to further reduce the computation time. Numerical experiments on the modeling of spatially homogeneous carbon dioxide vibrational relaxation were performed for optimized computational schemes of different orders. Based on them, the most optimal algorithm of calculations was recommended: a parallel EBDF scheme of fourth-order with an adaptive timestep. This method takes less computational time and memory costs and has the high stability.

Since their introduction in 1967, Lawson methods have achieved constant interest in the time discretization of evolution equations. The methods were originally devised for the numerical solution of stiff differential equations. Meanwhile, they constitute a well-established class of exponential integrators, which has turned out to be competitive for solving space discretizations of certain types of partial differential equations. The popularity of Lawson methods is in some contrast to the fact that they may have a bad convergence behaviour, since they do not satisfy any of the stiff order conditions. The aim of this paper is to explain this discrepancy. It is shown that non-stiff order conditions together with appropriate regularity assumptions imply high-order convergence of Lawson methods. Note, however, that the term regularity here includes the behaviour of the solution at the boundary. For instance, Lawson methods will behave well in the case of periodic boundary conditions, but they will show a dramatic order reduction for, e.g., Dirichlet boundary conditions. The precise regularity assumptions required for high-order convergence are worked out in this paper and related to the corresponding assumptions for splitting schemes. In contrast to previous work, the analysis is based on expansions of the exact and the numerical solution along the flow of the homogeneous problem. Numerical examples for the Schrödinger equation are included.

Hydraulic fracture propagation is a coupled solid-fluid problem involving moving boundaries. The boundary element method is an efficient way to model it in elastic media; however, the computational efficiency is still not promising because in most models the time-consuming implicit approach is used for iteratively solving the stiff equation of the targeted problem. In this paper, we present an explicit super-time-stepping algorithm based on the Runge–Kutta–Legendre method, to simulate a planar 3D hydraulic fracture propagation through the intermediately previous solutions. Each super-time-step covers s stages of single explicit time-step and the results generated can be around s2 times more stable than that using a single explicit time-step. Moreover, an adaptive super-time-step scheme is proposed to capture the marching fracture fronts based on the fracture growth rate. The modeling results are validated against analytical solutions for a penny-shaped hydraulic fracture and experimental results considering stress contrasts. Compared to the implicit iteration approach for the stiff equation, the explicit super-time-stepping algorithm can be up to 30 times faster because of no iteration and super-time-step. Finally, numerical examples are examined for fracture growth in rock formations with stress contrasts and anisotropic toughness, and all the results indicate that the explicit super-time-stepping algorithm can be a promising alternative to the implicit method.

In this paper, a seven-step block method for the solution of first order initial value problem in ordinary differential equations based on collocation of the differential equation and interpolation of the approximate solution using power series have been formed. The method is found to be consistent and zero-stable which guarantees convergence. Finally, numerical examples are presented to illustrate the accuracy and effectiveness of the method. Keywords: Power series, Collocation, Interpolation, Block method, Stiff.

In this paper, a numerical and analytical study was carried out in order to develop a new initial stiffness equation for thin-walled steel plate in bearing component. In this regard, effects of different design variables and curling occurrence on initial stiffness of this component were investigated. A set of parametric studies was performed on experimentally validated finite element models restraining curling occurrence to investigate the effect of geometrical parameters on initial stiffness of thin-walled steel plates. These design parameters included plate thickness, end distance, and edge distance and number of bolts. Effect of curling occurrence on initial stiffness was also considered using an analytical study. Finally, a new initial stiffness equation was proposed and compared with Eurocode3 design standard equation of hot-rolled steel plates. This new equation can be used in component-based modeling to calculate initial stiffness of thin-walled steel plate in bearing component. The outputs obtained by new equation showed better agreement with validated finite element results than those of old equation.

This article investigates the bending response performances of the magneto–electro–elastic (MEE) laminated plates resting on the Winkler foundation or the elastic half-space subjected to a transverse mechanical loading .By assuming that the foundation is not electrically and magnetically conductive, the scaled boundary finite element method (SBFEM) based on the three-dimensional (3D) theory of elasticity is applied for both the simulation of the MEE laminated plate and the elastic half-space. The SBFEM model considers the generalized displacement involving the elastic displacement, electric potential and magnetic potential as the nodal degree of freedom for the MEE laminated plates, and only the in-plane of the MEE laminated plate or the boundary of the elastic half-space needs to be discretized leading to reduce the spatial dimension by one. Furthermore, in the SBFEM, the governing equations can be solved by using an analytical approach in the radial direction of the scaled coordinate system, so that it is particularly suitable for the simulation of the elastic half-space. For the Winkler foundation–plate system, the global stiffness coupling governing equation that includes the interaction between the MEE laminated plate and the Winkler foundation is derived directly from the 3D elasticity equations of the MEE laminated plate by assuming that the foundation reactions are proportional to the transverse displacements of the plate structure. While for the MEE laminated plate-half-space system, the whole domain is divided into three sub-domains including the MEE laminated plate structure, the near and semi-infinite far foundation systems based on the sub-structure method, and then the stiffness matrix of each sub-domain can be determined by means of the SBFEM. As a result, the global stiffness equation of the plate-half-space system can be assembled according to the principle of the degree of freedom matching at the same nodes. The numerical results obtained for limiting cases by using the proposed method were compared with the published works and showed excellent agreements with the solutions based on the analytical and numerical approaches, so that the accuracy and applicability of the proposed formulations for the analysis of the interaction problems between the MEE laminated plate and the Winkler foundation or elastic half-space can be verified. Moreover, several numerical examples with various material properties, geometries, stacking sequences, aspect ratios, and supported boundary conditions were presented to show the effects of which on the responses of the plate–foundation system.

This paper discusses the analytical elastostatic stiffness modeling of parallel manipulators (PMs) considering the compliance of the link and joint. The proposed modeling is implemented in three steps: (1) the limb constraint wrenches are formulated based on screw theory; (2) the strain energy of the link and the joint is formulated using material mechanics and a mapping matrix, respectively, and the concentrated limb stiffness matrix corresponding to the constraint wrenches is obtained by summing the strain energy of the links and joints in the limb; and (3) the overall stiffness matrix is assembled based on the deformation compatibility equations. The strain energy factor index (SEFI) is adopted to describe the influence of the elastic components on the stiffness performance of the mechanism. Matrix structural analysis (MSA) using Timoshenko beam elements is applied to obtain analytical expressions for the compliance matrices of different joints through a three-step process: (1) formulate the element stiffness equation for each element; (2) extend the element stiffness equation to obtain the element contribution matrix, allowing the extended overall stiffness matrix to be obtained by summing the element contribution matrices; and (3) determine the stiffness matrices of joints by extracting the node stiffness matrix from the extended overall stiffness matrix and then releasing the degrees of freedom of twist. A comparison with MSA using Euler–Bernoulli beam elements demonstrates the superiority of using Timoshenko beam elements. The 2PRU-UPR PM is presented to illustrate the effectiveness of the proposed approach. Finally, the global SEFI and scatter matrix are used to identify the elastic component with the weakest stiffness performance, providing a new approach for effectively improving the stiffness performance of the mechanism.

A multi-step single-stage method is considered, which allows one to integrate stiff differential equations and systems of equations with high accuracy and low computational costs. The examples show that the proposed method is in solving stiff problems as good as the best available methods. The calculation results allow us to determine the absolute stability domains for the multi-step single-stage method, where it is possible to vary integration step within a wide range while maintaining the computational stability of the method.

Numerical modeling of nonequilibrium state-to-state carbon dioxide kinetics is a challenging time-consuming computational task that involves solving a huge system of stiff differential equations and requires optimized methods to solve it. In the present study, we propose and investigate optimizations for the extended backward differential formula (EBDF) scheme. Using adaptive timesteps instead of fixed ones reduces the number of steps in the algorithm many thousands of times, although with an increase in step complexity. The use of parallel computations to calculate relaxation terms allows one to further reduce the computation time. Numerical experiments on the modeling of spatially homogeneous carbon dioxide vibrational relaxation were performed for optimized computational schemes of different orders. Based on them, the most optimal algorithm of calculations was recommended: a parallel EBDF-scheme of fourth-order with an adaptive timestep. This method takes less computational time and memory costs and has the high stability.

Mathematical modelling of transient states of continuous stirred tank reactors was considered for the case of homogeneous liquid-phase reactions. A simple and very precise kinetic measurement strategy was proposed: registration of the reactor temperature during the start-up of the reaction process. The dynamic reactor model consisted of mass and energy balances written in dimensionless forms to enable general conclusions and easy computer implementation. Model simulations were carried out by using a numerical solver for stiff differential equations and a reaction invariant was revealed by theoretical analysis. A series of transient laboratory-scale experiments for a strongly exothermic reaction, a reaction between sodium thiosulphate and hydrogen peroxide was conducted and the kinetic parameters, the pre-exponential factors and activation energies were estimated. The data fitting was further improved by incorporating the decomposition of hydrogen peroxide as a side reaction. The estimated kinetic parameters were investigated further by sensitivity analysis and Markov-Chain-Monte-Carlo-Method (MCMC) to confirm their reliability. The proposed method is applicable for relatively simple systems with a measurable heat effect. For complex multireaction systems the method should be completed with chemical analysis.

We consider a standard heterogeneous agent model (HAM) that is widely used to analyze price developments in financial markets. The model is linear in log-prices and, in its basic setting, populated by fundamentalists and chartists. As the number of fundamentalists increases and exceeds a specific threshold, oscillations occur whose amplitude might even grow exponentially over time. From an economic perspective to adequately interpret such instability results it is indispensable to ensure that the characteristics and specific building blocks of the HAM are not at odds with the underlying structure of financial markets, in particular the specific trading rules. We expect that in markets with (almost) only fundamentalist traders prices might in the most extreme case oscillate, but never explode. In addition, if limit orders are available, prices should converge monotonically. Finally, if price bubbles occur in financial markets with fundamentalist traders, they should only result from the interactions between fundamentalists and the other traders, e.g., chartists, but not from fundamentalists’ decisions alone. From a mathematical perspective we show that the instability result common to the standard approach can be related to a “hidden” explicit discretization of a stiff ordinary differential equation contained in the model. Replacing this explicit discretization by an implicit one improves the model as it removes this artifact, bringing the model’s prediction in line with standard theory. The refined model still allows for price overshoots, bubbles, and crashes. However, in the implicit model these instabilities are caused by chartists and not by an unintended artifact.

An efficient finite-rate chemistry (FRC) model is developed for a preconditioned compressible flow solver. The model uses a point-implicit stiff ODE solver and a correlated dynamic adaptive chemistry algorithm. With respect to the conventional FRC model using the double precision variable coefficient stiff ODE solver, the present work achieves an 8.6 times speed-up for chemistry calculation, and 6.4 times for total computation, when using a 20-species kinetics mechanism for methane/air flames. As an example problem, a piloted partially premixed methane/air jet flame (Sandia Flame D), with a relatively low level of local extinction and re-ignition, is considered, and both the new FRC-large eddy simulation (LES) and flamelet/progress-variable (FPV)-LES are conducted. The FRC-LES approach predicts larger time-averaged flame length, and better agrees with the measured value. This is because the instantaneous high-temperature zone for the FPV-LES case is significantly smaller than it's FRC-LES counterpart, especially in the downstream region. For spatial distribution of time-averaged statistics, the FPV-LES result agrees with the experimental data better. For conditional statistics in the mixture fraction space, the FRC-LES approach provides significantly better predictions. Near the stoichiometric region, in comparison with experimental data and the FRC-LES results, the FPV-LES approach predicts higher radical generation, but lower CO generation and heat release.

This paper is concerned with the dynamic analysis of truss with initial member length imperfection under impulsive load considering geometric nonlinearity. Using displacement-based finite element formulation in solving the nonlinear problem of this truss requires incorporating the initial member length imperfection as a dependent boundary constraint to the master stiffness equation and producing a modified system of equations. For escaping the mathematical difficulties of treating the initial member length imperfection this paper proposes a novel approach to formulate the nonlinear vibration problem based on mixed finite element formulation. The dynamic equilibrium equation containing unknown displacements and forces is obtained using the principle of stationary potential energy. A mixed matrix of truss elements is established based on mixed variational formulation with length imperfection conditions considering nonlinear deformation. Combining the Newmark integration method and Newton Raphson iteration method is employed to solve the dynamic equations with geometric nonlinearity. Based on the employed method, the research develops the incremental-iterative algorithm and the calculation program for determining the dynamic response of truss with initial member length imperfection under impulsive load. The numerical results are presented to verify the efficiency of the proposed method.

The paper compares two ODE solvers using an example of a heat transfer equation. The sequential version of Brayton-Gustavson-Hatchel solver has been slightly inferior to Gear-Nordsieck solver. Algorithms profiling has led to the decision of parallelizing linear equation solving section and function evaluation. The first approach (parallelizing linear equations) improves performance of both algorithms. Second approach (parallelizing function evaluation) boosts BGH solver performance. Finally, it has been proved that wholly parallel version of BGH solver is more efficient with respect to processing time.

We study fluid distributions endowed with hyperbolical symmetry, which share many common features with Lemaitre-Tolman-Bondi (LTB) solutions (e.g. they are geodesic, shearing, non--conformally flat and the energy density is inhomogeneous). As such they may be considered as hyperbolically symmetric versions of LTB, with spherical symmetry replaced by hyperbolical symmetry. We start by considering pure dust models, and afterwards we extend our analysis to dissipative models with anisotropic pressure. In the former case the complexity factor is necessarily non-vanishing, whereas in the latter cases models with vanishing complexity factor are found. The remarkable fact is that all solutions satisfying the vanishing complexity condition are necessarily non-dissipative and satisfy the stiff equation of state., Comment: 11 pages Latex. Published in Entropy (Special Issue:Complexity of self-gravitating systems). arXiv admin note: text overlap with arXiv:2109.07758

Stiffness-based approach has been used to design the compliant version of mechanical iris. The rigid links of the mechanism are replaced by sheet flexures. Iris has found use as grippers and also to carry out controlled opening of aperture. This paper is based on formulating an estimate of the stiffness based on geometrical parameters. Stiffness equation relates the input moment to the output deflection. The equations are utilized to design based on the required use case.

The theory of Computational Singular Perturbation (CSP) has been widely used in the chemical kinetics and combustion community to understand complex chemistry, reduce chemical kinetic models and perform computational diagnostics in reacting flow simulations. Useful as it is, the application of CSP is nevertheless subject to computationally expensive eigen-decomposition of the chemical Jacobian matrix, especially for mechanisms involving a large number of species. In 2016–2017, a new CSP approach was developed by Lam, taking advantage of species lifetime and linearity in the governing equation of the fastest species. The corresponding quasi-state value of the current fastest species across the thin transition layer could be accurately predicted at each time step, during which period the concentration of the slower species is considered as constant. As a result, tedious explicit integration to resolve the apparent exponential transition layer, as those utilized in conventional implicit chemistry solvers (e.g., CVODE), is avoided and the integration time step could be largely extended. This method, referred to as linear CSP (LCSP), has been demonstrated to be useful in deriving reduced models and to numerically integrate stiff ODE systems. In 2018, an idea of nonlinear CSP (NCSP) following similar approach was further proposed by Lam, which abandons the enabling assumptions of the linearized governing equation for the fastest species and provides useful insights into the numerical integration of stiff chemical ODE systems. The development of CSP without expensive eigen-decomposition and computational evidence supporting its implementation are discussed and disseminated as a promising direction of advanced CSP theory. For general kinetic problems with complex couplings, a method based on sparse adaptive hybrid integration is introduced to demonstrate the redundancy issue and general stiffness in the fast subspace.

Although physical mechanism of earthquakes has not been clearly answered yet, it can be explained substantially with friction laws when the slip event subjects to the brittle crust. In such cases earthquakes are a result of frictional instability accompanied by stick-and-slip motion. Earthquakes, foreshocks, aftershocks, slow slip events have been modelled by using Rate-And-State Friction (RSF) laws. In this study Dieterich, Ruina and Perrin type RSF laws have been studied on the Burridge-Knopoff (BK) spring-block system, which was originally proposed with a velocity dependent friction law. In order to comply with the reality, fault geometry and its physical structure are chosen appropriate to the San Andreas/Parkfield fault. Since the proposed systems are stiff nonlinear dynamics, they are offered with numeric procedure adapted to solve stiff differential equations. By applying stability analysis, the critical boundaries between stable and unstable sliding (seismic cycle) are determined. The model is simulated by tuning the RSF law parameters for unstable sliding regime. As a result of the studies it has been found that, the magnitude of the slip event is proportional to the distance from the curve which separates stable and unstable sliding regimes. Besides, when system parameters deviates with a fixed amount from the stability curve, the system shows the same dynamics. To the best of my knowledge, the defined criterion is being published for the first time within the scope of this work. This study will pave the way for further researches of earthquake and weak triggering effects.

The implicit Runge–Kutta method with A-stability is suitable for solving stiff differential equations. However, the fully implicit Runge–Kutta method is very expensive in solving large system problems. Although some implicit Runge–Kutta methods can reduce the cost of computation, their accuracy and stability are also adversely affected. Therefore, an effective banded implicit Runge–Kutta method with high accuracy and high stability is proposed, which reduces the computation cost by changing the Jacobian matrix from a full coefficient matrix to a banded matrix. Numerical solutions and results of stiff equations obtained by the methods involved are compared, and the results show that the banded implicit Runge–Kutta method is advantageous to solve large stiff problems and conducive to the development of simulation.

Hybrid Trefftz Voronoi finite elements on polygon meshes are developed to capture thermal response of heterogeneous materials with randomly dispersed inclusions/voids. In this approach, the temperature intra-element field within the polygon is approximated by the homogeneous solutions to the heat conduction governing equation which are also called the T-complete functions. Whereas an auxiliary temperature frame field is independently defined on the polygonal boundary. The continuity at the matrix–inclusion interface is guaranteed by exterior and interior T-complete functions and that across element boundaries is enforced by incorporating the known hybrid functional at the element level. Thus, only the hybrid functional should be established for the matrix region. In conjunction with the divergence theorem, the domain integral involved in the functional vanishes, which leads to the element stiffness equation including boundary integrals only. It is reported that there exists a linear relationship between the maximum number of T-complete functions and the number of Gauss points sampled on each element side. Numerical results demonstrate that the proposed methodology is remarkably more efficient than the conventional finite element method (ABAQUS) and consequently becomes popular in micromechanical and microthermal modeling of composite and porous materials.

Thin-walled orthotropic shell structures are used in various fields of engineering, and the process of their deformation is essentially non-linear, which significantly complicates their analysis. A technique for numerically investigation of the process of deformation of such structures under dynamic loading is proposed in this article. The computational algorithm is based on the L.V. Kantorovich method and Rosenbrock method for solution of a stiff ODE system. A new variation of the mathematical model is based on the functional of total deformation energy, takes into account geometrical nonlinearity, transverse shears, orthotropy of the material, and rotational inertia, and may be used for a study of shells of different geometric forms. Dimensionless parameters are used. The proposed model and the use of mentioned methods allows to produce the most accurate calculations of the buckling of shell structures. The applicability of this technique is shown: the results of calculations are given for orthotropic shallow shells of double curvature having a square base with cylindrical and conical panels. The load values are derived for local and total stability loss, for structures made of four different materials. The effect of the rate of loading on the obtained values is shown.

Computational fluid dynamics (CFD) has become a major tool in understanding and predicting the behavior of reacting flows, which intrinsically involve the complex interaction of chemical kinetics and fluid mechanics. Among various open source CFD packages, the widely used C++ finite volume simulation toolbox OpenFOAM has many advantages such as its object-orientated framework, convenience to add multiphysics module and free availability, however, many weaknesses have been identified regarding its application in chemically reacting flows, especially incomplete splitting schemes, poor ordinary differential equation (ODE) solvers for stiff chemistry and the oversimplified mixture transport models, etc. In this work, an OpenFOAM 5.0 based reacting flow CFD platform reactingFoam-SCI is constructed by further implementing a few state-of-the-art advances in computational combustion, including a robust midpoint operator splitting scheme, and an accurate stable stiff ODE solver with error control. An interface between OpenFOAM 5.0 and the open source package Cantera 2.3.0 is also developed, to allow the evaluation of multicomponent and mixture-averaged transport properties and the call of other Cantera subroutines. The effectiveness and robustness of this new platform have been systematically validated against the analytical solutions and literature reported direct numerical calculation results, including those from a diffusion-convection problem, homogeneous ignition delay, shock tube, one-dimensional spherical flame initiation and propagation, two-dimensional unsteady premixed flame subject to hydrodynamic and diffusional-thermal instabilities, one-dimensional non-premixed counterflow flame and two-dimensional non-premixed co-flow flame. The satisfactory agreement demonstrates the accuracy and robustness of the current platform.

In this paper, two numerical methods are proposed for solving distributed-order fractional Bagley-Torvik equation. This equation is used in modeling the motion of a rigid plate immersed in a Newtonian fluid with respect to the nonnegative density function. Using the composite Boole 02BC &#xs rule the distributed-order Bagley-Torvik equation is approximated by a multi-term time-fractional equation, which is then solved by the Grunwald-Letnikov method &#x0028 GLM &#x0029 and the fractional differential transform method &#x0028 FDTM &#x0029. Finally, we compared our results with the exact results of some cases and show the excellent agreement between the approximate result and the exact solution.

A discontinuous spherically symmetric solution of the Einstein equations is found, containing an annihilation shock wave. Before the shock wave, there occurs a parabolic (with zero speed at infinity) compression of dust consisting of particles and antiparticles, which is glued on the wave of complete annihilation of antiparticles with hyperbolic expansion of a gas with an extremely stiff equation of state. Specifically, in contrast to the Friedmann homogeneous expansion. the solution behind the shock wave has a nonzero pressure gradient associated with acceleration of the continuum. Estimates are given for the parameters related to the stage of baryon-antibaryon annihilation at formation of the Universe.

A hybrid fundamental solution based finite element method (HFS-FEM) is proposed to analyze axisymmetric potential problems with arbitrary boundary conditions. The axisymmetric geometry is simplified from the three-dimensional (3D) to the two-dimensional (2D) by expanding boundary conditions into the summation of Fourier series. In the proposed approach, the interior potential field is constructed by utilizing a linear combination of fundamental solutions at source points as intra-element trial functions. And the frame potential field is independently introduced to enforce the continuity between adjacent elements. And then the two assumed fields are expanded into a series of symmetric and asymmetric components as done for the boundary conditions. For each component, the element stiffness equation involving boundary integrals only is established by means of the axisymmetric form of Hellinger-Reissner functional. Finally, the superposition principle is employed for the final solution. To assess the performance of HFS-FEM, three numerical examples are investigated and comparisons are conducted between the proposed approach and ABAQUS. The results show that the HFS-FEM exhibits insensitivity to mesh distortion.

International audience; Multipoint kinetic equation systems have been solved numerically using matrix algebra software and 1st to 4th order implicit schemes based on single-step matrix propagators. These matrix schemes have been validated successfully on demanding point kinetics benchmarks with various prescribed reactivity insertions. Verification tests have also been performed on a simple 3-region fast reactor core model with asymmetric step or ramp reactivity insertions and cross-checked with the multipoint kinetics SACRE code developed at INL and based on existing and validated stiff ODE solver packages. Accurate results can be obtained at a limited time expense. The reason for intriguing results obtained in some transients has been elucidated and linked to the multipoint matrix coefficient interpolation method during the transient.

Neutron transport has been still under very active development in research institutions and academia throughout the world. The spatial, temporal, energy and directional angle dependence make it remains one of the most computationally challenging problems in the world. The correct P 1 approximation to the neutron transport equation is not the first order diffusion equation, but the second-order in time, which is called telegraph equation. In this paper, a synopsis derivation of the point kinetics telegraph model is obtained from the neutron transport equation as a couple system of stiff differential equations. The problem of the obtained system is formulated in the matrix form and solved by higher orders Magnus expansion, where the first successive three orders of Magnus expansion are derived analytically for the point telegraph equations with multi-group of delayed neutrons and various different types of reactivity. These approximations depend on the exponential function of Magnus matrix, where the calculations are obtained using the eigenvalues of Magnus matrix and the corresponding eigenvectors. The proposed methods are tested using different cases of reactivity such as step, ramp and sinusoidal reactivity insertions. The numerical results indicate that the high order of Magnus expansion approximations is accurate compared with the traditional methods.

Ordinary differential equations (ODEs) are among the most important mathe-matical tools used in producing models in the physical sciences, biosciences, chemical sciences, engineering and many more fields. This has motivated re-searchers to provide efficient numerical methods for solving such equations. Most of these types of differential models are stiff, and suitable numerical methods have to be used to simulate the solutions. This paper starts with a sur-vey on the basic properties of stiff differential equations. Thereafter, we present the explicit one-step algorithm proposed by Fatunla to solve stiff systems of first-order scalar ODEs. As an illustrative example, we consider the Robertson problem (RP) which is known to be stiff. The results obtained with the explicit Fatunla method (EFM) are compared with those computed by the solver RADAU which is based on implicit Runge-Kutta methods. Our results are in good agreement with the latter ones.

Splitting methods are frequently used for large scale chemical reaction systems. The main reason is the reduced computational cost for solving the subproblems in comparison to the time integration of the full problem. However, a splitting error is introduced. The most popular splitting schemes are the first order Lie–Trotter splitting and the second order Strang splitting. However, in case of stiff differential equations the Strang splitting suffers from order reduction and both schemes have order one for stiff differential equations. Therefore, the step size restrictions due to the low order can result in a prohibitive small step size. Hence, splitting schemes with order larger than one are necessary for stiff differential equations. The Richardson extrapolation of the Lie–Trotter splitting is a second order scheme. In this paper we examine differential equations with a fast chemical source term and a slow transport term. Thus, all stiffness of the system is related to the chemical source term. We show that the Richardson extrapolation does not suffer from order reduction in this case. Furthermore, we perform a stability analysis of the extrapolated splitting scheme for a linear test problem. Thereby the operators of the test problem do not commute.

International audience; Real-time simulation is important for the fuel cell online diagnostics and hardware-in-the-loop (HIL) tests before industrial applications. However, it is hard to implement real-time multi-dimensional, multi-physical fuel cell models due to the model numerical stiffness issues. In this paper, the numerical stiffness of a tubular solid oxide fuel cell (SOFC) real-time model is first analyzed to identify the perturbation ranges related to the fuel cell electrochemical, fluidic and thermal domains. Some of the commonly used ordinary differential equation (ODE) solvers are then tested for the real-time simulation purpose. At last, a novel two-stage third-order parallel stiff ODE solver is proposed to improve the stability and reduce the multi-dimensional real-time fuel cell model execution time. To verify the proposed model and the ODE solver, real-time simulation experiments are carried out in a common embedded real-time platform. The experimental results show that the execution speed satisfies the requirement of real-time simulation. The solver stability under strong stiffness and the high model accuracy are also validated. The proposed real-time fuel cell model and the stiff ODE solver can also help to design the online diagnostic control method.

A new numerical method is presented for the solution of initial value problems described by systems of N linear ordinary differential equations (ODEs). Using the state-space representation, a differential equation of order n > 1 is transformed into a system of L = n×N first-order equations, thus the numerical method developed recently by Katsikadelis for first-order parabolic differential equations can be applied. The stability condition of the numerical scheme is derived and is investigated using several well-corroborated examples, which demonstrate also its convergence and accuracy. The method is simply implemented. It is accurate and has no numerical damping. The stability does not require symmetrical and positive definite coefficient matrices. This advantage is important because the scheme can find the solution of differential equations resulting from methods in which the space discretization does not result in symmetrical matrices, for example, the boundary element method. It captures the periodic behavior of the solution, where many of the standard numerical methods may fail or are highly inaccurate. The present method also solves equations having variable coefficients as well as non-linear ones. It performs well when motions of long duration are considered, and it can be employed for the integration of stiff differential equations as well as equations exhibiting softening where widely used methods may not be effective. The presented examples demonstrate the efficiency and accuracy of the method.

The guide stiffness performance directly affects the motion of the micromechanism in accuracy and security. Therefore, it is crucial to analyze the guide stiffness precisely. In this paper, a high-precision guide stiffness analysis method for the micromechanism by the boundary element method (BEM) is proposed. The validity and accuracy of the analysis method are tested by a guide stiffness experiment. In order to ensure the accuracy and safety during the micromechanism motion, a guiding unit of the micromechanism was designed based on the guiding principle. The guiding unit can provide parasitic motion and additional force in the motion of the micromechanism. Then, the stiffness equations of the beam element are derived by the boundary element method. The stiffness equation of straight circular flexure hinge is analyzed by rigid discretization and rigid combination, and the guide stiffness of the mechanism is investigated by rigid combination. Finally, according to the actual situation, the stiffness matrix of the guide rail (Kb) was proposed, and the analytical value of the guide stiffness was calculated to be 22.2 N/μm. The guide stiffness performance experiment was completed, and the experimental value is 22.3 N/μm. Therefore, the error between the analysis method and the experimental results is 0.45%. This study provides a new method for the stiffness analysis of high-precision micromechanisms and presents a reference for the design and stiffness analysis of complex structures. This method is helpful for stiffness analysis of the microrotary mechanism with high accuracy.

This chapter treats the numerical approximation of solutions of first-order differential equations, including stiff differential equations. The methods discussed and analyzed are the Euler method, the backward Euler method, the improved Euler method, and various fourth- and fifth-order Runge–Kutta algorithms. The concepts of local or cumulative truncation errors and propagated errors are discussed throughout.

Novel implications on neutron stars come from extended gravity. Specifically, the GW190814 event indicated the probability of having large mass stars in the mass-gap region $2.5-5\, M_{\odot}$. If the secondary component of GW190814 is a neutron star, such large masses are marginally supported by General Relativity (GR), since a very stiff Equation of State (EoS) would be needed to describe such large mass neutron stars, which would be incompatible with the GW170817 event, without any modification of gravity. In view of the two groundbreaking gravitational wave observations, we critically discuss the elevated role of extensions of GR towards the successful description of the GW190814 event, and we also speculate in a quantitative way on the important issue of the largest allowed neutron star mass., Comment: EPL Accepted

Delay differential equations (DDEs) are commonly used in pharmacometric models to describe delays present in pharmacokinetic and pharmacodynamic data analysis. Several DDE solvers have been implemented in NONMEM 7.5 for the first time. Two of them are based on algorithms already applied elsewhere, while others are extensions of existing ordinary differential equations (ODEs) solvers. The purpose of this tutorial is to introduce basic concepts underlying DDE based models and to show how they can be developed using NONMEM. The examples include previously published DDE models such as logistic growth, tumor growth inhibition, indirect response with precursor pool, rheumatoid arthritis, and erythropoiesis-stimulating agents. We evaluated the accuracy of NONMEM DDE solvers, their ability to handle stiff problems, and their performance in parameter estimation using both first-order conditional estimation (FOCE) and the expectation–maximization (EM) method. NONMEM control streams and excerpts from datasets are provided for all discussed examples. All DDE solvers provide accurate and precise solutions with the number of significant digits controlled by the error tolerance parameters. For estimation of population parameters, the EM method is more stable than FOCE regardless of the DDE solver. published

This paper focuses on the treatment of nonlinear multi freedom constraints using an augmented Lagrangian method in finite element analysis of frames. The process of imposing boundary constraints is developed by changing the assembly stiffness equation to produce a modified system of equation considering nonlinear multi freedom constraints. For imposing the nonlinear constraints two better methods are the penalty augmentation method and Lagrange multiplier adjunction method. But there are not free of disadvantages. Using penalty method has a disadvantage in the choice appropriate weight values that balance solution accuracy with the violation of constraint conditions. Using the Lagrange multiplier adjunction method requires additional unknowns, and more complicated storage allocation procedures. This research proposes the connection between these methods using the augmented Lagrangian method for imposing the nonlinear multi freedom constraints in finite element analysis of frame. Based on the Newton Raphson method the incremental-iterative algorithm for solving the nonlinear balanced equations is established.

The article introduces a new multistage technique for solving a polynomial system of nonlinear initial and boundary value problems of differential equations. The radius of convergence R of the series solution to the problem is derived a-priorly in terms of the parameters of the polynomial system. Then guided by the convergence-control parameter h, Mathematics; Multistage Parker-Sochacki Method; Approximate analytical method; Epidemic model; SIR model; Combustion equation; Lorenz problem; Chaotic problem; Troesch's problem