Computational models for multi-scale coupled dynamic problems

Melnik, Roderick V. N.; Roberts, Amanda

doi:10.1016/j.future.2003.07.009

Cited by 14 publications

(10 citation statements)

References 24 publications

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“…Monte Carlo procedures and Markov chain-based approximations have become some of the major tools in addressing complex problems involving uncertainties in a number of application areas [22][23][24][25][26]. Many such problems are multiscale in nature [27][28][29][30][31][32][33][34][35]. In a number of practically important cases, such multiscale problems require the development of mathematical models that go beyond the description of MULTIVARIATE JUMP-DIFFUSION PROCESSES the same parameters and the discretization size of time period = 0.0002.…”

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confidence: 99%

First passage time for multivariate jump‐diffusion processes in finance and other areas of applications

Zhang

Melnik

2008

Appl Stoch Models Bus & Ind

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The first passage time (FPT) problem is an important problem with a wide range of applications in science, engineering, economics, and industry. Mathematically, such a problem can be reduced to estimating the probability of a stochastic process first to reach a boundary level. In most important applications in the financial industry, the FPT problem does not have an analytical solution and the development of efficient numerical methods becomes the only practical avenue for its solution. Most of our examples in this contribution are centered around the evaluation of default correlations in credit risk analysis, where we are concerned with the joint defaults of several correlated firms, the task that is reducible to a FPT problem. This task represents a great challenge for jump-diffusion processes (JDP). In this contribution, we develop further our previous fast Monte Carlo method in the case of multivariate (and correlated) JDP. This generalization allows us, among other things, to evaluate the default events of several correlated assets based on a set of empirical data. The developed technique is an efficient tool for a number of financial, economic, and business applications, such as credit analysis, barrier option pricing, macroeconomic dynamics, and the evaluation of risk, as well as for a number of other areas of applications in science and engineering, where the FPT problem arises.Many applications in science, engineering, finance, economics, and industry require the solution of the first passage problem. Examples range from population genetics and other applications in biology, to neurobiology and neurophysiology, to molecule kinetics and problems in chemistry and physics, to rare events modelling and quantum information, and to aerospace engineering applications (e.g. [1][2][3][4]). This problem also arises in financial applications. Although methodologies developed in this paper are applicable to a number of other areas, we exemplify our discussion here with problems from finance.In the financial world, individual companies are usually linked together via economic conditions, so default correlation, defined as the risk of multiple companies' default together, has been an important area of research in credit analysis with applications to joint defaults, credit derivatives, asset pricing, and risk management.Currently, there are two dominant groups of theoretical models used in default correlation. One is a reduced form model, such as in [5] that uses a Copula function to parameterize the default correlation. In this context, it is worthwhile noting that Chen and Sopranzetti [6] have translated the joint default probability into a bivariate normal probability function, making the analysis and applications of such models potentially more convenient.The second group of models for default correlation is a structural form model. Zhou [7] and Hull and White [8] were the first to incorporate default correlation into the Black-Cox first passage structural model. They obtained similar closed-form solutions for two ...

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confidence: 99%

First passage time for multivariate jump‐diffusion processes in finance and other areas of applications

Zhang

Melnik

2008

Appl Stoch Models Bus & Ind

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“…In the multidimensional case, the developed approach for dynamic models of SMA and associated coupled multiscale problems [52,53] were put on a systematic basis via the centre-manifold theory and received its justification in a series of papers [54][55][56]. Twodimensional thermo-mechanical waves in SMA patches were first analyzed in [57] and in [58] distributed mechanical loadings for patches of different sizes were studied.…”

Section: Dynamic Models and Their Approximationsmentioning

confidence: 99%

Nonlinear dynamics of shape memory alloy oscillators in tuning structural vibration frequencies

Wang

Melnik

2012

Mechatronics

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“…where the peak piezomagnetic coefficientd =d cr + a∆σ + b(∆σ) 2 , overstress ∆σ = σ − σ cr , z = |H|/H,H =H cr + ζ∆σ, A = σ cr /σ, σ is the stress, ε is the strain and σ cr is the critical inherent stress of domain switching. a, b,d cr ,H cr and ζ are material constants obtained by relating the Gibbs free energy and fitting experimental data as discussed in [14].…”

Section: Constitutive Model Based On Density Of Domain Switching (Dds)mentioning

confidence: 99%

“…In recent time, multiscale modeling and computation have become powerful tool in solving many problems related to strongly coupled dynamics and multiphysics phenomena in material science and engineering [1,2]. Such an attempt, with judicious choice of the mathematical models which represent the mutually inseparable physical processes at the molecular scale, the lattice scale, the microstructural scale, and at the continuum scale, can be highly reliable in interpreting the experimental results and also in designing new materials and devices with desired multifunctionality.…”

Section: Introductionmentioning

confidence: 99%

“…In context of time-dependent problems, the concept of consistent temporal integration [12] is worth noting, which extend itself beyond the classical notion of upwinding. Subsequently, while attempting a strongly nonlinear and coupled problem, the advantage in adopting a low-dimensional manifold based on appropriate error estimate during iteration [2,13] may be noted. In the present paper we report a new variationally consistent multiscale finite model with the following two features: (1) a sub-grid scale for upwinding the miscrostructural information and (2) an approximate manifold of the original system that retains the desired accuracy in the finite element solution, which is unlike the usual ordering scheme as followed in the conventional asymptotic method of truncation.…”

Section: Introductionmentioning

confidence: 99%

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Multiscale Finite Element Modeling of the Coupled Nonlinear Dynamics of Magnetostrictive Composite Thin Film

Mahapatra

Ghosh

Gopalakrishnan

2005

Lecture Notes in Computer Science

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Abstract.A multiscale nonlinear finite element model for analysis and design of the deformation gradient and the magnetic field distribution in Terfenol-D/epoxy thin film device under Transverse Magnetic (TM) mode of operation is developed in this work. A phenomenological constitutive model based on the density of domain switching (DDS) of an ellipsoidal inclusion in unit cell of matrix is implemented. A sub-grid scale homogenization technique is employed to upwind the microstructural information. A general procedure to ensure the solution convergence toward an approximate inertial manifold is reported.

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Computational models for multi-scale coupled dynamic problems

Cited by 14 publications

References 24 publications

First passage time for multivariate jump‐diffusion processes in finance and other areas of applications

First passage time for multivariate jump‐diffusion processes in finance and other areas of applications

Nonlinear dynamics of shape memory alloy oscillators in tuning structural vibration frequencies

Multiscale Finite Element Modeling of the Coupled Nonlinear Dynamics of Magnetostrictive Composite Thin Film

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