Relevant bibliographies by topics / Numerically stiff

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Numerically stiff'

Author: Grafiati
Published: 6 September 2023

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Journal articles on the topic "Numerically stiff"

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Piché, R., and A. Ellman. "Numerical Integration of Fluid Power Circuit Models Using Two-Stage Semi-Implicit Runge-Kutta Methods." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 208, no. 3 (May 1994): 167–75. http://dx.doi.org/10.1243/pime_proc_1994_208_114_02.

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Fluid power circuits that contain fluid volumes of different orders of magnitude are difficult to simulate because the system of ordinary differential equations is numerically stiff. Even algorithms specially designed for stiff systems require excessively small time steps to avoid numerical oscillation in simulations of some circuits. In this paper the accuracy and numerical stability of several two-stage semi-implicit Runge-Kutta methods that have been proposed in circuit simulation literature are analysed and compared. It is shown that, for integration of very stiff circuits, the best method in this class is an L-stable method. A simple numerical example is used to verify the theoretical results. The example includes a novel way of modelling orifice flow that is especially suitable for numerical simulations.
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Dear, J., Z. Shi, and J. Lin. "An efficient numerical integration system for stiff unified constitutive equations for metal forming applications." IOP Conference Series: Materials Science and Engineering 1270, no. 1 (December 1, 2022): 012008. http://dx.doi.org/10.1088/1757-899x/1270/1/012008.

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Unified constitutive equations have been developed in recent years to predict viscoplastic flow and microstructural evolution of metal alloys for metal forming applications. These equations can be implemented into commercial FE code, such as ABAQUS and PAMSTAMP, to predict mechanical and physical properties of materials in a wide range of metal forming processes. These equations are normally stiff and need significant computer CPU time to solve. In this research, a series of numerical analyses are performed to investigate the difficulties within MATLAB of solving these stiff unified constitutive equations. A metric is introduced to allow evaluation of the numerical stiffness to assess the most appropriate numerical integration method. This metric is based on the ratio of maximum to minimum eigenvalue. This metric allows for an appropriate numerical method to be chosen giving more effective modelling of deformation and plasticity processes. Based on the theoretical work described above, a user-friendly system, based on MATLAB, is then developed for numerically integrating these types of stiff constitutive equations. This is particularly useful for metal forming engineers and researchers who need an effective computational tool to determine constitutive properties well based on numerical integration theories.
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Asnor, Mohd Yatim, and Ibrahim. "Solving Directly Higher Order Ordinary Differential Equations by Using Variable Order Block Backward Differentiation Formulae." Symmetry 11, no. 10 (October 14, 2019): 1289. http://dx.doi.org/10.3390/sym11101289.

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Variable order block backward differentiation formulae (VOHOBBDF) method is employedfor treating numerically higher order Ordinary Differential Equations (ODEs). In this respect, the purpose of this research is to treat initial value problem (IVP) of higher order stiff ODEs directly. BBDF method is symmetrical to BDF method but it has the advantage of producing more than one solutions simultaneously. Order three, four, and five of VOHOBBDF are developed and implemented as a single code by applying adaptive order approach to enhance the computational efficiency. This approach enables the selection of the least computed LTE among the three orders of VOHOBBDF and switch the code to the method that produces the least LTE for the next step. A few numerical experiments on the focused problem were performed to investigate the numerical efficiency of implementing VOHOBBDF methods in a single code. The analysis of the experimental results reveals the numerical efficiency of this approach as it yielded better performances with less computational effort when compared with built-in stiff Matlab codes. The superior performances demonstrated by the application of adaptive orders selection in a single code thus indicate its reliability as a direct solver for higher order stiff ODEs.
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Braileanu, G. "Matrix operators for numerically stable representation of stiff linear dynamic systems." IEEE Transactions on Automatic Control 35, no. 8 (1990): 974–80. http://dx.doi.org/10.1109/9.58516.

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Solovarova, Liubov S., and Ta D. Phuong. "On the numerical solution of second-order stiff linear differential-algebraic equations." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 24, no. 2 (June 30, 2022): 151–61. http://dx.doi.org/10.15507/2079-6900.24.202202.151-161.

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This article addresses systems of linear ordinary differential equations with an identically degenerate matrix in the main part. Such formulations of problems in literature are usually called differential-algebraic equations. In this work, attention is paid to the problems of the second order. Basing on the theory of matrix pencils and polynomials, sufficient conditions for existence and uniqueness of the equations’ solution are given. To solve them numerically, authors investigate a multistep method and its version based on a reformulated notation of the original problem. This representation makes it possible to construct methods whose coefficient matrices can be calculated at previous points. This approach has delivered good results in numerical solution of first-order differential-algebraic equations that contain stiff and rapidly oscillating components and have singular matrix pencil. The stability of proposed numerical algorithm is investigated for the well-known test equation. It is shown that this difference scheme has the first order of convergence. Numerical calculations of the model problem are presented.
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Gao, Pan, Zhihui Liu, Ji Zeng, Yiting Zhan, and Fei Wang. "A Random Forest Model for the Prediction of Spudcan Penetration Resistance in Stiff-Over-Soft Clays." Polish Maritime Research 27, no. 4 (December 1, 2020): 130–38. http://dx.doi.org/10.2478/pomr-2020-0073.

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Abstract Punch-through is a major threat to the jack-up unit, especially at well sites with layered stiff-over-soft clays. A model is proposed to predict the spudcan penetration resistance in stiff-over-soft clays, based on the random forest (RF) method. The RF model was trained and tested with numerical simulation results obtained through the Finite Element model, implemented with the Coupled Eulerian Lagrangian (CEL) approach. With the proposed CEL model, the effects of the stiff layer thickness, undrained shear strength ratio, and the undrained shear strength of the soft layer on the bearing characteristics, as well as the soil failure mechanism, were numerically studied. A simplified resistance profile model of penetration in stiff-over-soft clays is proposed, divided into three sections by the peak point and the transition point. The importance of soil parameters to the penetration resistance was analysed. Then, the trained RF model was tested against the test set, showing a good prediction of the numerical cases. Finally, the trained RF was validated against centrifuge tests. The RF model successfully captured the punch-through potential, and was verified using data recorded in the field, showing advantages over the SNAME guideline. It is supposed that the trained RF model should give a good prediction of the spudcan penetration resistance profile, especially if trained with more field data.
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Grenestedt, Joachim L., and Mikael Danielsson. "Elastic-Plastic Wrinkling of Sandwich Panels With Layered Cores." Journal of Applied Mechanics 72, no. 2 (March 1, 2005): 276–81. http://dx.doi.org/10.1115/1.1828063.

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Elastic-plastic wrinkling of compression loaded sandwich panels made with layered cores was studied analytically and experimentally. A core with a stiff layer near the sandwich skins can improve various properties, including wrinkling and impact strengths, with only a minor weight penalty. The 2D plane stress and plane strain bifurcation problems were solved analytically, save for a determinantal equation which was solved numerically. Experiments were performed on aluminum skin/foam core sandwich panels with different combinations of stiff and soft core materials. Good correlation between experiments and theory was obtained.
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Chen, Shanqin. "Krylov SSP Integrating Factor Runge–Kutta WENO Methods." Mathematics 9, no. 13 (June 24, 2021): 1483. http://dx.doi.org/10.3390/math9131483.

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Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.
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Albi, Giacomo, Young-Pil Choi, and Axel-Stefan Häck. "Pressureless Euler alignment system with control." Mathematical Models and Methods in Applied Sciences 28, no. 09 (August 2018): 1635–64. http://dx.doi.org/10.1142/s0218202518400018.

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We study a non-local hydrodynamic system with control. First, we characterize the control dynamics as a sub-optimal approximation to the optimal control problem constrained to the evolution of the pressureless Euler alignment system. We then discuss the critical thresholds that lead to global regularity or finite-time blow-up of strong solutions in one and two dimensions. Finally, we use a finite volume scheme, coupled with an implicit–explicit time integrator to solve numerically the stiff scale of the controlled system. Several numerical simulations are shown to validate the theoretical and computational results of the paper.
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Tudor, M. "A test of numerical instability and stiffness in the parametrizations of the ARPÉGE and ALADIN models." Geoscientific Model Development 6, no. 4 (July 5, 2013): 901–13. http://dx.doi.org/10.5194/gmd-6-901-2013.

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Abstract. Meteorological numerical weather prediction (NWP) models solve a system of partial differential equations in time and space. Semi-lagrangian advection schemes allow for long time steps. These longer time steps can result in instabilities occurring in the model physics. A system of differential equations in which some solution components decay more rapidly than others is stiff. In this case it is stability rather than accuracy that restricts the time step. The vertical diffusion parametrization can cause fast non-meteorological oscillations around the slowly evolving true solution (fibrillations). These are treated with an anti-fibrillation scheme, but small oscillations remain in operational weather forecasts using ARPÉGE and ALADIN models. In this paper, a simple test is designed to reveal if the formulation of particular a physical parametrization is a stiff problem or potentially numerically unstable in combination with any other part of the model. When the test is applied to a stable scheme, the solution remains stable. However, applying the test to a potentially unstable scheme yields a solution with fibrillations of substantial amplitude. The parametrizations of the NWP model ARPÉGE were tested one by one to see which one may be the source of unstable model behaviour. The test identified the set of equations in the stratiform precipitation scheme (a diagnostic Kessler-type scheme) as a stiff problem, particularly the combination of terms arising due to the evaporation of snow.
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Dissertations / Theses on the topic "Numerically stiff"

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Ashi, Hala. "Numerical methods for stiff systems." Thesis, University of Nottingham, 2008. http://eprints.nottingham.ac.uk/10663/.

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Some real-world applications involve situations where different physical phenomena acting on very different time scales occur simultaneously. The partial differential equations (PDEs) governing such situations are categorized as "stiff" PDEs. Stiffness is a challenging property of differential equations (DEs) that prevents conventional explicit numerical integrators from handling a problem efficiently. For such cases, stability (rather than accuracy) requirements dictate the choice of time step size to be very small. Considerable effort in coping with stiffness has gone into developing time-discretization methods to overcome many of the constraints of the conventional methods. Recently, there has been a renewed interest in exponential integrators that have emerged as a viable alternative for dealing effectively with stiffness of DEs. Our attention has been focused on the explicit Exponential Time Differencing (ETD) integrators that are designed to solve stiff semi-linear problems. Semi-linear PDEs can be split into a linear part, which contains the stiffest part of the dynamics of the problem, and a nonlinear part, which varies more slowly than the linear part. The ETD methods solve the linear part exactly, and then explicitly approximate the remaining part by polynomial approximations. The first aspect of this project involves an analytical examination of the methods' stability properties in order to present the advantage of these methods in overcoming the stability constraints. Furthermore, we discuss the numerical difficulties in approximating the ETD coefficients, which are functions of the linear term of the PDE. We address ourselves to describing various algorithms for approximating the coefficients, analyze their performance and their computational cost, and weigh their advantages for an efficient implementation of the ETD methods. The second aspect is to perform a variety of numerical experiments to evaluate the usefulness of the ETD methods, compared to other competing stiff integrators, for integrating real application problems. The problems considered include the Kuramoto-Sivashinsky equation, the nonlinear Schrödinger equation and the nonlinear Thin Film equation, all in one space dimension. The main properties tested are accuracy, start-up overhead cost and overall computation cost, since these parameters play key roles in the overall efficiency of the methods.
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Addenbrooke, Trevor Ian. "Numerical analysis of tunnelling in stiff clay." Thesis, Online version, 1996. http://ethos.bl.uk/OrderDetails.do?did=1&uin=uk.bl.ethos.243326.

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Ingram, Peter James. "The application of numerical models to natural stiff clays." Thesis, City University London, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.340454.

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Lee, Gordon Tsz Kit. "Three-dimensional numerical studies of "NATM" tunnelling in stiff clay /." View Abstract or Full-Text, 2003. http://library.ust.hk/cgi/db/thesis.pl?CIVL%202003%20LEE.

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Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2003.
Includes bibliographical references (leaves 202-209). Also available in electronic version. Access restricted to campus users.
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Summersgill, Freya. "Numerical modelling of stiff clay cut slopes with nonlocal strain regularisation." Thesis, Imperial College London, 2014. http://hdl.handle.net/10044/1/34567.

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The aim of this project is to investigate the stability of cut slopes in stiff clay. The findings are subsequently applied to model stabilisation with piles, used to remediate failure of existing slopes and stabilise potentially unstable slopes created by widening transport corridors. Stiff clay is a strain softening material, meaning that soil strength reduces as the material is strained, for example in the formation of a slip surface. In an excavated slope this can lead to a progressive, brittle slope failure. Simulation of strain softening behaviour is therefore an important aspect to model. The interaction of piles and stiff clay cut slopes is investigated using the Imperial College Geotechnics section's finite element program ICFEP. In designing a suitable layout of the finite element mesh, preliminary analyses found the two existing local strain softening models to be very dependent on the size and arrangement of elements. To mitigate this shortcoming, a nonlocal strain softening model was implemented in ICFEP. This model controls the development of strain by relating the surrounding strains to the calculation of strain at that point, using a weighting function. Three variations of the nonlocal formulation are evaluated in terms of their mesh dependence. A parametric study with simple shear and biaxial compression analyses evaluated the new parameters required by the nonlocal strain softening model. The nonlocal results demonstrated very low mesh dependence and a clear improvement on the local strain softening models. In order to examine the mesh dependence of the new model in a boundary value problem compared to the local strain softening approach, excavated slope analyses without piles were first performed. The slope was modelled in plane strain with coupled consolidation. These analyses also investigated other factors such as the impact of adopting a small strain stiffness material model on the development of the failure mechanism and the impact of the spatial variation of permeability on the time to failure. The final set of analyses constructed vertical stabilisation piles in the excavated slope, represented as either solid elements or one dimensional beam elements. The development of various failure mechanisms for stiff clay cuttings was found to be dependent on pile location, pile diameter and pile length. This project provides an insight into the constitutive model and boundary conditions required to study stabilisation piles in a stiff clay cutting. The nonlocal model performed very well to reduce mesh dependence, confirming the biaxial compression results. However, the use of coupled consolidation was found to cause further mesh dependence of the results.
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Tanner, Gregory Mark. "Generalized additive Runge-Kutta methods for stiff odes." Diss., University of Iowa, 2018. https://ir.uiowa.edu/etd/6507.

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In many applications, ordinary differential equations can be additively partitioned \[y'=f(y)=\sum_{m=1}^{N}\f{}{m}(y).] It can be advantageous to discriminate between the different parts of the right-hand side according to stiffness, nonlinearity, evaluation cost, etc. In 2015, Sandu and G\"{u}nther \cite{sandu2015gark} introduced Generalized Additive Runge-Kutta (GARK) methods which are given by \begin{eqnarray*} Y_{i}^{\{q\}} & = & y_{n}+h\sum_{m=1}^{N}\sum_{j=1}^{s^{\{m\}}}a_{i,j}^{\{q,m\}}f^{\{m\}}\left(Y_{j}^{\{m\}}\right)\\ & & \text{for } i=1,\dots,s^{\{q\}},\,q=1,\dots,N\\ y_{n+1} & = & y_{n}+h\sum_{m=1}^{N}\sum_{j=1}^{s^{\{m\}}}b_{j}^{\{m\}}f^{\{m\}}\left(Y_{j}^{\{m\}}\right)\end{eqnarray*} with the corresponding generalized Butcher tableau \[\begin{array}{c|ccc} \c{}{1} & \A{1,1} & \cdots & \A{1,N}\\\vdots & \vdots & \ddots & \vdots\\ \c{}{N} & \A{N,1} & \cdots & \A{N,N}\\\hline & \b{}{1} & \cdots & \b{}{N}\end{array}\] The diagonal blocks $\left(\A{q,q},\b{}{q},\c{}{q}\right)$ can be chosen for example from standard Runge-Kutta methods, and the off-diagonal blocks $\A{q,m},\:q\neq m,$ act as coupling coefficients between the underlying methods. The case when $N=2$ and both diagonal blocks are implicit methods (IMIM) is examined. This thesis presents order conditions and simplifying assumptions that can be used to choose the off-diagonal coupling blocks for IMIM methods. Error analysis is performed for stiff problems of the form \begin{eqnarray*}\dot{y} & = & f(y,z)\\ \epsilon\dot{z} & = & g(y,z)\end{eqnarray*} with small stiffness parameter $\epsilon.$ As $\epsilon\to 0,$ the problem reduces to an index 1 differential algebraic equation provided $g_{z}(y,z)$ is invertible in a neighborhood of the solution. A tree theory is developed for IMIM methods applied to the reduced problem. Numerical results will be presented for several IMIM methods applied to the Van der Pol equation.
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Nguyen, Thi Hoai Thuong. "Numerical approximation of boundary conditions and stiff source terms in hyperbolic equations." Thesis, Rennes 1, 2020. http://www.theses.fr/2020REN1S027.

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Ce travail est consacré à l’étude théorique et numérique de systèmes hyperboliques d’équations aux dérivées partielles et aux équations de transport, avec des termes de relaxation et des conditions aux bords. Dans la première partie, on étudie la stabilité raide d’approximations numériques par différences finies du problème mixte donnée initiale-donnée au bord pour l’équation des ondes amorties dans le quart de plan. Dans le cadre du schéma discret en espace, nous proposons deux méthodes de discrétisation de la condition de Dirichlet. La première est la technique de sommation par partie et la seconde est basée sur le concept de condition au bord transparente. Nous proposons également une comparaison numérique des deux méthodes, en particulier de leur domaine de stabilité. La deuxième partie traite de schémas numériques d’ordre élevé pour l’équation de transport avec une donnée entrante sur domaine borné. Nous construisons, implémentons et analysons la procédure de Lax-Wendroff inverse au bord entrant. Nous obtenons des taux de convergence optimaux en combinant des estimations de stabilité précises pour l’extrapolation des conditions au bord avec des développements de couche limite numérique. Dans la dernière partie, nous étudions la stabilité de solutions stationnaires pour des systèmes non conservatifs avec des termes géométrique et de relaxation. Nous démontrons que les solutions stationnaires sont stables parmi les solutions entropique processus, qui généralisent le concept de solutions entropiques faibles. Nous supposons essentiellement que le système est complété par une entropie partiellement convexe et que, selon la dissipation du terme de relaxation, la stabilité ou la stabilité asymptotique des solutions stationnaires est obtenue
The dissertation focuses on the study of the theoretical and numerical analysis of hyperbolic systems of partial differential equations and transport equations, with relaxation terms and boundary conditions. In the first part, we consider the stiff stability for numerical approximations by finite differences of the initial boundary value problem for the linear damped wave equation in a quarter plane. Within the framework of the difference scheme in space, we propose two methods of discretization of Dirichlet boundary condition. The first is the technique of summation by part and the second is based on the concept of transparent boundary conditions. We also provide a numerical comparison of the two numerical methods, in particular in terms of stability domain. The second part is about high order numerical schemes for transport equations with nonzero incoming boundary data on bounded domains. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at incoming boundary. We obtain optimal convergence rates by combining sharp stability estimate for extrapolation boundary conditions with numerical boundary layer expansions. In the last part, we study the stability of stationary solutions for non-conservative systems with geometric and relaxation source term. We prove that stationary solutions are stable among entropy process solution, which is a generalisation of the concept of entropy weak solutions. We mainly assume that the system is endowed with a partially convex entropy and, according to the entropy dissipation provided by the relaxation term, stability or asymptotic stability of stationary solutions is obtained
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Montanelli, Hadrien. "Numerical algorithms for differential equations with periodicity." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:cc001282-4285-4ca2-ad06-31787b540c61.

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This thesis presents new numerical methods for solving differential equations with periodicity. Spectral methods for solving linear and nonlinear ODEs, linear ODE eigenvalue problems and linear time-dependent PDEs on a periodic interval are reviewed, and a novel approach for computing multiplication matrices is presented. Choreographies, periodic solutions of the n-body problem that share a common orbit, are computed for the first time to high accuracy using an algorithm based on approximation by trigonometric polynomials and optimization techniques with exact gradient and exact Hessian matrix. New choreographies in spaces of constant curvature are found. Exponential integrators for solving periodic semilinear stiff PDEs in 1D, 2D and 3D periodic domains are reviewed, and 30 exponential integrators are compared on 11 PDEs. It is shown that the complicated fifth-, sixth- and seventh-order methods do not really outperform one of the simplest exponential integrators, the fourth-order ETDRK4 scheme of Cox and Matthews. Finally, algorithms for solving semilinear stiff PDEs on the sphere with spectral accuracy in space and fourth-order accuracy in time are proposed. These are based on a new variant of the double Fourier sphere method in coefficient space and standard implicit-explicit time-stepping schemes. A comparison is made against exponential integrators and it is found that implicit-explicit schemes perform better. The algorithms described in each chapter of this thesis have been implemented in MATLAB and made available as part of Chebfun.
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Yang, Lei. "Fracture Behaviour of Layered Rocks with Alternating Stiff and Soft Layers." Thesis, The University of Sydney, 2022. https://hdl.handle.net/2123/29608.

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Various subsurface engineering activities, including the stimulation of unconventional hydrocarbon reservoirs, the development of geothermal energy and the drilling and blasting operations, have been increasingly carried out in sedimentary rocks with layering structures. The success of these activities is reliant on the formation of fracture network created by hydraulic fracturing or the extension and interconnection of fractures to break the layered rock. This thesis dedicates to the fracture behaviours of layered rock with alternating stiff and soft layers. First, a new damage-plasticity constitutive model which takes account of the effect of confining pressure and strain rate on the strength and post-peak behaviour is proposed for layered rocks’ components (e.g., stiff layers, soft layers, and layer interfaces) subjected to various loading scenarios. The robustness and accuracy of the new model are demonstrated by validating against available experimental results and by benchmarking with the reported simulations. Then, the new constitutive model is used to numerically explore the fracture evolution behaviour of layered rock discs in the Brazilian test. The effects of inclination angle, Young's modulus of layer interface and mechanical contrast ratio on the fracture mechanism of layered rock disc are investigated. Finally, a versatile hydromechanical coupled finite-discrete element method is employed to simulate the non-planar three-dimensional simultaneous growth of multiple hydraulic fractures in layered tight reservoirs with various mechanical contrast ratios. The mechanism behind the simultaneous growth and methods to promote the simultaneous growth are also discussed. The numerical results obtained from this thesis provide some guidelines in designing the engineering projects conducted in layered rocks and thus help field operators to maximize the productivity.
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Tallarek, Ulrich. "Electrokinetic flow and transport in porous media: Experimental methods, numerical analysis, and applications." [S.l. : s.n.], 2004. http://deposit.ddb.de/cgi-bin/dokserv?idn=974460923.

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Books on the topic "Numerically stiff"

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LeVeque, Randall J. A study of numerical methods for hyperbolic conservation laws with stiff source terms. Moffett Field, Calif: National Aeronautics and Space Administration, Ames Research Center, 1988.

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The numerical solution of nonlinear stiff initial value problems: An analysis of one step methods. Amsterdam: Centrum voor Wiskunde en Informatica, 1985.

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Center, Langley Research, ed. A new theory for multistep discretizations of stiff ordinary differential equations: Stability with large step sizes. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1985.

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Enenkel, Robert Frederick. DIMSEMs--Diagonally IMplicit Single-Eigenvalue Methods for the numerical solution of stiff ordinary differential equations on parallel computers. Toronto: University of Toronto, Dept. of Computer Science, 1996.

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1946-, Verwer J. G., ed. Numerical solution of time-dependent advection-diffusion-reaction equations. Berlin: Springer, 2003.

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Quantum and semi-classical percolation and breakdown in disordered solids. Berlin: Springer-Verlag, 2009.

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National Aeronautics and Space Administration (NASA) Staff. Study of Numerical Methods for Hyperbolic Conservation Laws with Stiff Source Terms. Independently Published, 2018.

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Wanner, Gerhard, and E. Hairer. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (Springer Series in Computational Mathematics). 2nd ed. Springer, 2004.

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Wanner, Gerhard, and E. Hairer. Solving Ordinary Differential Equations II: Stiff and Differential - Algebraic Problems. Springer London, Limited, 2013.

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Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, 2010.

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Book chapters on the topic "Numerically stiff"

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Savcenco, V., and R. M. M. Mattheij. "Multirate Numerical Integration for Stiff ODEs." In Progress in Industrial Mathematics at ECMI 2008, 327–32. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-12110-4_50.

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Rauber, Thomas, and Gudula Rünger. "Parallel Solution of Stiff Ordinary Differential Equations." In Parallel Numerical Computation with Applications, 33–51. Boston, MA: Springer US, 1999. http://dx.doi.org/10.1007/978-1-4615-5205-5_3.

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Abdulle, Assyr. "Explicit Methods for Stiff Stochastic Differential Equations." In Numerical Analysis of Multiscale Computations, 1–22. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-21943-6_1.

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Lam, S. H. "Singular Perturbation for Stiff Equations Using Numerical Methods." In Recent Advances in the Aerospace Sciences, 3–19. Boston, MA: Springer US, 1985. http://dx.doi.org/10.1007/978-1-4684-4298-4_1.

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Wang, Shan Yong, K. C. Lam, Ivan W. H. Fung, Wan Cheng Zhu, Tao Xu, and Lian Chong Li. "Numerical Study of Crack Propagation in Stiff Clays." In Fracture and Damage Mechanics V, 201–4. Stafa: Trans Tech Publications Ltd., 2006. http://dx.doi.org/10.4028/0-87849-413-8.201.

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Atanasova, Pavlina Kh, Stefani A. Panayotova, Elena V. Zemlyanaya, Yury M. Shukrinov, and Ilhom R. Rahmonov. "Numerical Simulation of the Stiff System of Equations Within the Spintronic Model." In Numerical Methods and Applications, 301–8. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-10692-8_33.

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Hongyuan, Fu, and Chen Guannan. "Numerical Computation of Stiff Systems for Nonequilibrium Ionization Problems." In Large Scale Scientific Computing, 75–82. Boston, MA: Birkhäuser Boston, 1987. http://dx.doi.org/10.1007/978-1-4684-6754-3_5.

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Šmarda, Zdeněk. "Numerical Solving Stiff Control Problems for Delay Differential Equations." In Recent Advances in Soft Computing, 299–310. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-97888-8_27.

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Zhelyazov, Todor, and Sergey Pshenichnov. "Simulation of the Mechanical Wave Propagation in a Viscoelastic Media With and Without Stiff Inclusions." In Numerical Methods and Applications, 339–48. Cham: Springer Nature Switzerland, 2023. http://dx.doi.org/10.1007/978-3-031-32412-3_30.

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Coulier, P., A. Dijckmans, J. Jiang, D. J. Thompson, G. Degrande, and G. Lombaert. "Stiff Wave Barriers for the Mitigation of Railway Induced Vibrations." In Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 539–46. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-662-44832-8_63.

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Conference papers on the topic "Numerically stiff"

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Stojanoski, Goran, Dimitar Ninevski, Gerhard Rath, and Matthew Harker. "Multidimensional Trajectory Tracking for Numerically Stiff Independent Metering System." In SICFP’21 The 17:th Scandinavian International Conference on Fluid Power. Linköping University Electronic Press, 2021. http://dx.doi.org/10.3384/ecp182p283.

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This paper presents a new approach for solving an optimal control problem in a hydraulic system, using a variational calculus method. It uses a path tracking method of two different states with different units and of different magnitude. To ensure the uniqueness of the solution, two regularization terms were introduced, whose influence is regulated by regularization parameters. The system of differential equations, obtained from the Euler-Lagrange equations of the variational problem, was solved by a mass matrix method and discretized with linear differential operators at the interstitial points for numerical stability. This enabled the calculation of the control variables, despite the stiffness of the numerical problem. The results obtained show an energy-efficient performance and no oscillations. Finally, a Simulink model of the hydraulic system was created in which the calculated control variables were inserted as feed-forward inputs, to verify the results.
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Ahn, H. "An implicit method for numerically stiff venting problems in honeycomb and other multicell configurations." In Fluid Dynamics Conference. Reston, Virigina: American Institute of Aeronautics and Astronautics, 1994. http://dx.doi.org/10.2514/6.1994-2361.

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Esque´, Salvador, Asko Ellman, and Robert Piche´. "Numerical Integration of Pressure Build-Up Volumes Using an L-Stable Rosenbrock Method." In ASME 2002 International Mechanical Engineering Congress and Exposition. ASMEDC, 2002. http://dx.doi.org/10.1115/imece2002-39343.

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Simulation of fluid power systems has become a tool widely used for testing, designing and virtual prototyping. The choice of a numerical integration method for solving stiff systems of ordinary differential equations is a key factor for achieving proper stability and computational efficiency during simulation. Whereas widely used A-stable methods require small integration step sizes in order to avoid numerical oscillations when solving numerically stiff problems, the L-stable Rosenbrock method presented in this paper can take large steps. The method is implemented with an estimator of the local truncation error and a predictor of the step size. Simulations results show the good performance of the integrator in terms of both stability and efficiency.
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Boston, D. Matthew, Jose R. Rivas-Padilla, and Andres F. Arrieta. "Design and Manufacturing of a Multi-Stable Selectively Stiff Morphing Section Demonstrator." In ASME 2019 Conference on Smart Materials, Adaptive Structures and Intelligent Systems. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/smasis2019-5706.

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Abstract Morphing wings offer potential efficiency and performance benefits for aircraft fulfilling multiple mission requirements. However, the design of shape adaptable wings is limited by the inherent design trade-offs of weight, aerodynamic control authority, and load-carrying capacity. A potential solution to this trilemma is proposed by exploiting the stiffness adaptability of thin, curved structures which geometric instability results in two statically stable states. We design and manufacture a morphing wing section demonstrator composed of two compliant 3D printed ribs monolithically embedded with the proposed bi-stable elements. The demonstrator’s structural response is numerically modelled and compared with experimental results from a static loading test. A deflection field of the response under mechanical actuation is obtained through digital image correlation. Numerical and experimental results indicate the capability of the wing section to achieve four distinct stable configurations with varying global stiffness behavior.
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Stojanoski, Goran, Dimitar Ninevski, Gerhard Rath, and Matthew Harker. "A Novel Method for Solving an Optimal Control Problem for a Numerically Stiff Independent Metering System." In 2020 Australian and New Zealand Control Conference (ANZCC). IEEE, 2020. http://dx.doi.org/10.1109/anzcc50923.2020.9318391.

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Malysheva, Julia, and Heikki Handroos. "Fast Calculation of Stiff Hydraulic Models Using the Modified Pseudo-Dynamic Solver." In BATH/ASME 2020 Symposium on Fluid Power and Motion Control. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/fpmc2020-2805.

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Abstract The work addresses the problem of the fast and accurate calculation of the mathematically stiff hydraulic models using the modified pseudo-dynamic solver (PDS). In particular, it studies which of the numerical integration methods inside the modified PDS ensure efficient calculation of the stiff hydraulic model. In the work, the operating principle of the modified PDS is described. The effect of the three different fixed-step integration methods (Euler, Runge-Kutta of fourth order, and modified Heun’s method) are considered. The numerical stability of the modified Heun’s method is improved by substituting the purely turbulent orifice model with the two-regime orifice model. The two-regime orifice accounts for both the turbulent and laminar flows and thus allows to avoid the numerical problems related to the small pressure drops. As a numerical example of the mathematically stiff hydraulic model a hydraulic circuit with the two-way flow control valve which contains small volume is employed. As the implementation environment for the developed simulation models the compiled C language that supports the real-time simulation is chosen. The solutions obtained for the numerical example using the modified PDS based on the three integration methods, their accuracies and calculation speeds are presented in comparison with the solution obtained using conventional integration procedure. The obtained results show that, in general, the modified PDS allows to solve numerically stiff hydraulic models in a very efficient way ensuring accelerated simulation with the high solution accuracy. It is also shown that the simulation speed-up can be obtained not only by the complexity reduction of the numerical integration method employed inside the modified PDS but also by increasing its numerical stability.
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Fujikawa, Takeshi, and Etsujiro Imanishi. "A Precise and Stiffly Stable Time Integration Method for Vibration Equations." In ASME 2001 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/detc2001/vib-21320.

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Abstract A method of time integration algorithm is presented for solving stiff vibration and motion problems. It is absolutely stable, numerically dissipative, and much accurate than other dissipative time integration methods. It achieves high-frequency dissipation, while minimizing unwanted low-frequency dissipation. In this method change of acceleration during time step is expressed as quadratic function including some parameters, whose appropriate values are determined through numerical investigation. Two calculation examples are demonstrated to show the usefulness of this method.
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Changizi, M. Amin, and Ion Stiharu. "A Complete Parametric Study of Pull-In Voltage by Nonlinear Differential Equation." In ASME 2014 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/imece2014-37744.

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Micro-cantilever beams are interested structures in MEMS because of their fabrication is very easy and its versatility. The importance of micro-cantilevers beam in MEMS has driven various investigations like static and dynamic performances under different loading such as potential fields. In this research the non-linear differential equation which models dynamics of a micro-cantilever beams vibration subjected to electrostatic field has been studied. The model which has one degree of freedom is used to calculate the pull-in voltage. This model adopted based on different method of calculating stiffness of micro-cantilever beam. The nonlinear ordinary differential equation which used to model the dynamics of the cantilever subjected to electric field close to snap on is highly stiff. Investigation on solving of nonlinear stiff ordinary equation showed that only Lsode algorithm yield to correct solution to the problem. Lsode is equipped with a robust adaptive time step selection mechanism that enables solutions to very stiff problems, as the one under discussion. The best match in the resonant frequency for equivalent stiffness based on four different models was considered. The stiffness model suitable for the best match in deflection is proved to be different from the model that yields. Pull-in voltage under electric field was studied. Pull-in voltage has been investigated from the analytical and numerical perspective. A complete parametric study of structural damping effect on large deflection of micro-cantilever beam was studied was done numerically in this work. Different kind of impulse voltages were considered and effect of them on pulling voltage numerically was studied. A cumbersome mathematical method, Lie symmetry, was used to drive a closed from of time response to step voltage for undamped system and pull in voltage of such system was calculated. Finally, a closed form driven from the nonlinear ODE for calculating pulling voltage was presented.
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Käppi, T. J., A. U. Ellman, and R. Piché. "Implementation of Rosenbrock Integration Algorithm With Adaptive Step Size Control in Time-Domain Simulation of Fluid Power Systems." In ASME 1998 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 1998. http://dx.doi.org/10.1115/imece1998-0468.

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Abstract Numerical simulation models of fluid power systems typically have a large scale of different time constants and so can be considered as numerically stiff systems. The Rosenbrock method is a differential equation integration algorithm with suitable accuracy and stability properties. In this paper the implementation of the Rosenbrock method as part of a time-domain fluid power system simulation package is presented. Adaptive time step size control makes it possible to achieve an optimal computational effort with controllable integration error based on user-defined error tolerance. Computation of examples show considerable improvement in accuracy and computational speed over the previously used integration method.
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Liermann, Matthias, Christian Feller, and Florian Lindinger. "Real-Time Simulation of Fluid Power Systems." In ASME/BATH 2021 Symposium on Fluid Power and Motion Control. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/fpmc2021-70304.

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Abstract System-simulations involving fluid-power structures often result in numerically stiff model equations which may require prohibitively small simulation time steps when being tackled with a fixed-step solver. This poses a challenge in situations where real-time performance is required. This paper presents a practical rule-of-thumb to estimate the maximum permissible step-size for a given fluid power system and explains the influence of the relevant physical quantities on the step size requirement in simple terms. A categorization of methods suitable to relax the step-size requirement is proposed. Many research papers have been produced about methods and examples of how to improve real-time performance of fluid power systems, or stiff systems in general. The proposed categorization can be seen as a map for the simulation engineer to understand the basic point-of-attacks for the real-time simulation problem.
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Reports on the topic "Numerically stiff"

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Tan, Peng, and Nicholas Sitar. Parallel Level-Set DEM (LS-DEM) Development and Application to the Study of Deformation and Flow of Granular Media. Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA, March 2023. http://dx.doi.org/10.55461/kmiz5819.

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We present a systematic investigation of computational approaches to the modeling of granular materials. Granular materials are ubiquitous in everyday life and in a variety of engineering and industrial applications. Despite the apparent simplicity of the laws governing particle-scale interactions, predicting the continuum mechanical response of granular materials still poses extraordinary challenges. This is largely due to the complex history dependence resulting from continuous rearrangement of the microstructure of granular material, as well as the mechanical interlocking due to grain morphology and surface roughness. X-Ray Computed Tomography (XRCT) is used to characterize the grain morphology and the fabric of the granular media, naturally deposited sand in this study. The Level-Set based Discrete Element Method (LS-DEM) is then used to bridge the granular behavior gap between the micro and macro scale. The LS-DEM establishes a one-to-one correspondence between granular objects and numerical avatars and captures the details of grain morphology and surface roughness. However, the high-fidelity representation significantly increases the demands on computational resources. To this end a parallel version of LS-DEM is introduced to significantly decrease the computational demands. The code employs a binning algorithm, which reduces the search complexity of contact detection from O(n2) to O(n), and a domain decomposition strategy is used to elicit parallel computing in a memory- and communication-efficient manner. The parallel implementation shows good scalability and efficiency. High fidelity LS avatars obtained from XRCT images of naturally deposited sand are then used to replicate the results of triaxial tests using the new, parallel LS-DEM code. The result show that both micro- and macro-mechanical behavior of natural material is well captured and is consistent with experimental data, confirming experimental observation that the primary source of peak strength of sand is the mechanical interlocking between irregularly shaped grains. Specifically, triaxial test simulations with a flexible membrane produce a very good match to experimentally observed relationships between deviatoric stress and mobilized friction angle for naturally deposited sand. We then explore the viability of modeling dynamic problems with a new formulation of an impulse based LS-DEM. The new formulation is stable, fast, and energy conservative. However, it can be numerically stiff when the assembly has substantial mass differences between particles. We also demonstrate the feasibility of modeling deformable structures in the rigid body framework and propose several enhancements to improve the convergence of collision resolution, including a hybrid time integration scheme to separately handle at rest contacts and dynamic collisions. Finally, we extend the impulse-based LS-DEM to include arbitrarily shaped topographic surfaces and exploit its algorithmic advantages to demonstrate the feasibility of modeling realistic behavior of granular flows. The novel formulation significantly improves performance of dynamic simulations by allowing larger time steps, which is advantageous for observing the full development of physical phenomena such as rock avalanches, which we present as an illustrative example.
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Werner, L., and F. Odeh. Numerical Methods for Stiff Ordinary and Elliptic Partial Differential Equations. Fort Belvoir, VA: Defense Technical Information Center, February 1985. http://dx.doi.org/10.21236/ada153247.

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Walker, H. F. Numerical solution of nonlinear algebraic equations in stiff ODE solving (1986--89)---Quasi-Newton updating for large scale nonlinear systems (1989--90). Office of Scientific and Technical Information (OSTI), January 1990. http://dx.doi.org/10.2172/6132932.

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Walker, H. F. Numerical solution of nonlinear algebraic equations in stiff ODE solving (1986--89)---Quasi-Newton updating for large scale nonlinear systems (1989--90). Final report, 1986--1990. Office of Scientific and Technical Information (OSTI), December 1990. http://dx.doi.org/10.2172/10109632.

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Levesque, Justine, Nathaniel Loranger, Carter Sehn, Shantel Johnson, and Jordan Babando. COVID-19 prevalence and infection control measures at homeless shelters and hostels in high-income countries: protocol for a scoping review. York University Libraries, 2021. http://dx.doi.org/10.25071/10315/38513.

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The COVID-19 pandemic has disproportionately impacted people experiencing homelessness. Homeless shelters and hostels, as congregate living spaces for residents with many health vulnerabilities, are highly susceptible to outbreaks of COVID-19. A synthesis of the research-to-date can inform evidence-based practices for infection, prevention, and control strategies at these sites to reduce the prevalence of COVID-19 among both shelter/hostel residents and staff. Methods: A scoping review in accordance with Arksey and O’Malley’s framework will be conducted to identify literature reporting COVID-19 positivity rates among homeless shelter and hostel residents and staff, as well as infection control strategies to prevent outbreaks in these facilities. The focus will be on literature produced in high-income countries. Nine academic literature databases and 11 grey literature databases will be searched for literature from March 2020 to July 2021. Literature screening will be completed by two reviewers and facilitated by Covidence, a systematic review management platform. A third reviewer will be engaged to resolve disagreements and facilitate consensus. A narrative summary of the major themes identified in the literature, numerical counts of relevant data including the COVID-19 positivity rates, and recommendations for different infection control approaches will be produced. Discussion: The synthesis of the research generated on COVID-19 prevalence and prevention in homeless shelters and hostels will assist in establishing best practices to prevent the spread of COVID-19 and other airborne diseases at these facilities in high-income countries while identifying next steps to expand the existing evidence base.
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Mazzoni, Silvia, Nicholas Gregor, Linda Al Atik, Yousef Bozorgnia, David Welch, and Gregory Deierlein. Probabilistic Seismic Hazard Analysis and Selecting and Scaling of Ground-Motion Records (PEER-CEA Project). Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA, November 2020. http://dx.doi.org/10.55461/zjdn7385.

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This report is one of a series of reports documenting the methods and findings of a multi-year, multi-disciplinary project coordinated by the Pacific Earthquake Engineering Research Center (PEER) and funded by the California Earthquake Authority (CEA). The overall project is titled “Quantifying the Performance of Retrofit of Cripple Walls and Sill Anchorage in Single-Family Wood-Frame Buildings,” henceforth referred to as the “PEER–CEA Project.” The overall objective of the PEER–CEA Project is to provide scientifically based information (e.g., testing, analysis, and resulting loss models) that measure and assess the effectiveness of seismic retrofit to reduce the risk of damage and associated losses (repair costs) of wood-frame houses with cripple wall and sill anchorage deficiencies as well as retrofitted conditions that address those deficiencies. Tasks that support and inform the loss-modeling effort are: (1) collecting and summarizing existing information and results of previous research on the performance of wood-frame houses; (2) identifying construction features to characterize alternative variants of wood-frame houses; (3) characterizing earthquake hazard and ground motions at representative sites in California; (4) developing cyclic loading protocols and conducting laboratory tests of cripple wall panels, wood-frame wall subassemblies, and sill anchorages to measure and document their response (strength and stiffness) under cyclic loading; and (5) the computer modeling, simulations, and the development of loss models as informed by a workshop with claims adjustors. This report is a product of Working Group 3 (WG3), Task 3.1: Selecting and Scaling Ground-motion records. The objective of Task 3.1 is to provide suites of ground motions to be used by other working groups (WGs), especially Working Group 5: Analytical Modeling (WG5) for Simulation Studies. The ground motions used in the numerical simulations are intended to represent seismic hazard at the building site. The seismic hazard is dependent on the location of the site relative to seismic sources, the characteristics of the seismic sources in the region and the local soil conditions at the site. To achieve a proper representation of hazard across the State of California, ten sites were selected, and a site-specific probabilistic seismic hazard analysis (PSHA) was performed at each of these sites for both a soft soil (Vs30 = 270 m/sec) and a stiff soil (Vs30=760 m/sec). The PSHA used the UCERF3 seismic source model, which represents the latest seismic source model adopted by the USGS [2013] and NGA-West2 ground-motion models. The PSHA was carried out for structural periods ranging from 0.01 to 10 sec. At each site and soil class, the results from the PSHA—hazard curves, hazard deaggregation, and uniform-hazard spectra (UHS)—were extracted for a series of ten return periods, prescribed by WG5 and WG6, ranging from 15.5–2500 years. For each case (site, soil class, and return period), the UHS was used as the target spectrum for selection and modification of a suite of ground motions. Additionally, another set of target spectra based on “Conditional Spectra” (CS), which are more realistic than UHS, was developed [Baker and Lee 2018]. The Conditional Spectra are defined by the median (Conditional Mean Spectrum) and a period-dependent variance. A suite of at least 40 record pairs (horizontal) were selected and modified for each return period and target-spectrum type. Thus, for each ground-motion suite, 40 or more record pairs were selected using the deaggregation of the hazard, resulting in more than 200 record pairs per target-spectrum type at each site. The suites contained more than 40 records in case some were rejected by the modelers due to secondary characteristics; however, none were rejected, and the complete set was used. For the case of UHS as the target spectrum, the selected motions were modified (scaled) such that the average of the median spectrum (RotD50) [Boore 2010] of the ground-motion pairs follow the target spectrum closely within the period range of interest to the analysts. In communications with WG5 researchers, for ground-motion (time histories, or time series) selection and modification, a period range between 0.01–2.0 sec was selected for this specific application for the project. The duration metrics and pulse characteristics of the records were also used in the final selection of ground motions. The damping ratio for the PSHA and ground-motion target spectra was set to 5%, which is standard practice in engineering applications. For the cases where the CS was used as the target spectrum, the ground-motion suites were selected and scaled using a modified version of the conditional spectrum ground-motion selection tool (CS-GMS tool) developed by Baker and Lee [2018]. This tool selects and scales a suite of ground motions to meet both the median and the user-defined variability. This variability is defined by the relationship developed by Baker and Jayaram [2008]. The computation of CS requires a structural period for the conditional model. In collaboration with WG5 researchers, a conditioning period of 0.25 sec was selected as a representative of the fundamental mode of vibration of the buildings of interest in this study. Working Group 5 carried out a sensitivity analysis of using other conditioning periods, and the results and discussion of selection of conditioning period are reported in Section 4 of the WG5 PEER report entitled Technical Background Report for Structural Analysis and Performance Assessment. The WG3.1 report presents a summary of the selected sites, the seismic-source characterization model, and the ground-motion characterization model used in the PSHA, followed by selection and modification of suites of ground motions. The Record Sequence Number (RSN) and the associated scale factors are tabulated in the Appendices of this report, and the actual time-series files can be downloaded from the PEER Ground-motion database Portal (https://ngawest2.berkeley.edu/)(link is external).
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