Multi-time Scaling Research Articles

The General Analytic Expression of a Harvested Logistic Model with Slowly Varying Coefficients

The harvested logistic model with a slow variation in coefficients has been considered. Two cases, which depend on the harvest rate, were identified. The first one is when the harvest is subcritical, where the population evolves to an equilibrium. The other is supercritical harvesting, where the population decreases to zero at finite times. The single analytic approximate expression, which is capable of describing both harvesting cases, is readily and explicitly obtained using the multi-time scaling method together with the perturbation approach. This solution fits for a wide range of coefficient values. In addition, such an expression is validated by utilizing numerical computations, which are obtained by using the fourth-order Runge–Kutta technique. Finally, the comparison shows a very good agreement between the two methods.

A Slow Single-Species Model with Non-Symmetric Variation of the Coefficients

A single-species population model exhibiting a symmetric slow variation for the carrying capacity and intrinsic growth rate is evaluated explicitly. However, it is unrealistic to eliminate the possibility of a clear separation in the evolution of the biotic environmental elements; thus, this paper considers the situation where these elements have a hierarchical variation on the time scales. Accordingly, two particular situations are recognized, which are the carrying capacity varies faster than the growth rate and vice versa. Applying the multi-time scaling technique in such a system provides a small parameter, which leads us to construct analytical approximate expressions for the population behavior, using the perturbation approach. Such approximations display very good agreement with the numerical simulations.

WATMUS: Wavelet Transformation-Induced Multi-time Scaling for Accelerating Fatigue Simulations at Multiple Spatial Scales

WATMUS: Wavelet Transformation-Induced Multi-time Scaling for Accelerating Fatigue Simulations at Multiple Spatial Scales

Multi-time scale based modeling of piezoelectric materials coupling transient electrical and dynamic fields with finite deformation damage

This paper develops a high performance scalable computational model for simulating coupled transient electric and finite deformation dynamic fields with widely discrepant frequencies, governing the behavior of piezoelectric materials in the finite element framework. The nonlinear piezoelectric constitutive laws are formulated in the context of continuum damage models (CDM) at finite deformation. A fully coupled, total Lagrangian finite element formulation is developed for modeling the electric and mechanical fields in a staggered way. To address the computational challenge in modeling piezoelectric devices with large discrepancy in the frequenlightsghtscies of the electrical signal and mechanical vibrations, a wavelet transformation induced multi-time scaling (WATMUS) algorithm is developed. The WATMUS method projects the field with higher frequency onto the lower frequency problem through the use of orthogonal wavelet transformation. No assumption of periodic response is needed in the WATMUS scheme. While being accurate, the WATMUS algorithm significantly enhances the computational efficiency in comparison with single time scale integration methods. The accuracy and efficiency of the WATMUS algorithm are validated through piezoelectric applications.

Computational modeling of finite deformation piezoelectric material behavior coupling transient electrical and mechanical fields

This paper develops a finite element formulation and computational model for coupling electric and finite strain dynamic fields with widely discrepant frequencies, governing the behavior of piezoelectric materials. The piezoelectric materials are defined by time-dependent nonlinear constitutive laws. A fully coupled, total Lagrangian finite element formulation is developed for modeling the electric and mechanical fields. A challenge in computational modeling of piezoelectric devices arise due to large discrepancy in the frequencies of the electrical signal and mechanical vibrations, especially when a large number of mechanical cycles need to be simulated. A wavelet transformation induced multi-time scaling (WATMUS) algorithm is developed for dynamic piezoelectric simulations. The WATMUS algorithm projects the high frequency electric fields on to the lower frequency of displacement and velocity fields, on which time integration is performed. The method significantly enhances the computational efficiency in comparison with single time scale integration methods. The accuracy and efficiency of the WATMUS algorithm are validated through piezo-electric applications.

Multi-time scaling based modeling of transient electro-magnetic fields in vibrating media with antenna applications

This paper develops an accurate and efficient finite element model for simulating coupled transient electromagnetic and dynamic mechanical fields that differ widely in the frequency ranges. This coupled modeling framework is necessary for effective modeling and simulation of structures such as antennae that are governed by multi-physics problems operating in different frequency and temporal regimes. A key development is the wavelet transformation induced multi-time scaling or WATMUS method that is designed to overcome shortcomings of modeling coupled multi-physics problems that are governed by disparate frequencies. The WATMUS-based FE model is enhanced in this paper with a scaled and preconditioned Newton-GMRES solver for efficient solution. Results from the WATMUS-based FE model show the accuracy and highly improved computational efficiency in comparison with single time-scale methods. The coupled FE model is used to solve two different antenna problems with large electromagnetic to mechanical frequency ratios. The examples considered are a monopole antenna and a microstrip patch antenna. Comparing the electromagnetic fields with the progression of mechanical cycles demonstrate complex multi-physics relations in these applications.

Wavelet transformation induced multi-time scaling (WATMUS) model for coupled transient electro-magnetic and structural dynamics finite element analysis

Multi-functional devices that integrate electromagnetic and mechanical fields are gaining importance in a wide variety of applications. The combined mechanical and electromagnetic regimes, encompassed in these structures, make it necessary to develop effective multi-physics analysis tools at a range of temporal scales. An important consideration is that the different fields governing multi-physics response may have large frequency discrepancies, e.g. the ultra-high electromagnetic frequencies and moderate vibration frequencies. Computational analyses of these discrepant frequency problems using conventional time integration schemes can become intractable. This paper develops a framework for coupling transient electromagnetic and dynamic fields to predict the evolution of electric and magnetic fields and their fluxes in a vibrating substrate undergoing finite deformation. It addresses the issue of time integration with large frequency ratios, by introducing a novel wavelet transformation induced multi-time scaling (WATMUS) method in the finite element framework. The method significantly enhances the computational efficiency in comparison with conventional single time scale integration methods. An adaptive enhancement of WATMUS scheme allows for the optimal wavelet bases in the transformation and integration step sizes. The accuracy and efficiency of the proposed WATMUS scheme is verified by comparing the results with the single time-scale simulations of the coupled transient electromagnetic nonlinear vibration problems.

Accelerating cyclic plasticity simulations using an adaptive wavelet transformation based multitime scaling method

SUMMARYCyclic finite element simulations of complex materials, for example, polycrystalline metals, are widely used to study fatigue failure due to plasticity and damage. Typically, this requires the simulation of a large number of cycles to failure for accurate determination of evolving deformation variables. Modeling cyclic deformation using conventional methods of time integration in semidiscretization techniques can however be computationally challenging. Single time scale integration methods typically follow the high frequency characteristics and discretize each cycle into a number of time steps over which integration is performed. To overcome this computational challenge, the wavelet transformation‐based multitime scale (WATMUS) method proposed in an earlier work by the authors is advanced and validated in this paper to perform accelerated finite element simulations of materials undergoing rate‐dependent plasticity for large number of cycles. Specifically, the WATMUS algorithm is integrated with crystal plasticity finite element method to perform accelerated simulations of polycrystalline alloys. The WATMUS methodology is also endowed with adaptive capabilities to optimally construct the wavelet basis functions and determine coarse‐scale cycle steps. Accuracy and efficiency of the WATMUS methodology is conclusively demonstrated by comparing the results with cyclic single‐time scale crystal plasticity finite element simulations performed on image‐based microstructure of titanium alloys. Copyright © 2012 John Wiley & Sons, Ltd.

Microstructure and load sensitive fatigue crack nucleation in Ti-6242 using accelerated crystal plasticity FEM simulations

This paper investigates microstructure and load sensitive fatigue behavior of Ti-6242 using cyclic crystal plasticity finite element (CPFE) simulations of statistically equivalent image-based microstructures. A wavelet transformation induced multi-time scaling (WATMUS) method [1,2] is used to perform accelerated cyclic CPFE simulations till crack nucleation, otherwise infeasible using conventional time integration schemes. A physically motivated crack nucleation model in terms of crystal plasticity variables [3] is extended in this work to predict nucleation. The crack nucleation model is based on dislocation pile-up and stress concentration at grain boundaries, caused by inhomogeneous plastic deformation in the polycrystalline microstructure. The model is calibrated and validated with experiments. The dependence of yield strength on the underlying grain orientations and sizes is developed through the introduction of an effective microstructural parameter Plastic Flow Index or PFI. To determine the effects of the microstructure on crack nucleation, a local microstructural variable is defined in terms of the surface area fraction of soft grains surrounding each hard grain or SAFSSG. Simulations with different cyclic load patterns suggest that fatigue crack nucleation in Ti-6242 strongly depends on the dwell cycle hold time at maximum stress.

Wavelet transformation based multi-time scaling method for crystal plasticity FE simulations under cyclic loading

Microstructure based mechanistic calculations, coupled with physically motivated crack initiation criterion, can provide effective means to predict fatigue cracking in polycrystalline materials. However the accommodation of large number of cycles to failure, as observed in the experiments, could be computationally exhaustive to simulate using conventional single time scale finite element analysis. To meet this challenging requirement, a novel wavelet transformation based multi-time scaling algorithm is proposed for accelerated crystal plasticity finite element simulations in this paper. An advantage over other conventional methods that fail because of assumptions of periodicity etc., is that no assumption of scale separation is needed with this method. The wavelet decomposition naturally retains the high frequency response through the wavelet basis functions and transforms the low frequency material response into a “cycle scale” problem with monotonic evolution. The method significantly enhances the computational efficiency in comparison with conventional single time scale integration methods. Adaptivity conditions are also developed for this algorithm to improve accuracy and efficiency. Numerical examples for validating the multi-scaling algorithm are executed for a one dimensional viscoplastic problem and a 3D crystal plasticity model of polycrystalline Ti alloy under the cyclic loading conditions.

Numerical study of multitime scaling in a solid system undergoing phase separation.

The multitime-scaling rule for time correlation functions is shown in systems undergoing phase separation. With the use of multitime scaling, the asymptotic form of the structural function, S\ifmmode \tilde{}\else \~{}\fi{}(x)\ensuremath{\propto}${x}^{4}$, at small scaled wave numbers x is shown in a late state of phase separation in the diffusional system. A numerical simulation is performed to examine the multitime scaling. We use a discretized Cahn-Hilliard equation with the Oono-Puri free energy. The chemical potential is not conserved at all and may satisfy multitime scaling in the late stage of phase separation. Hence the suggested ${x}^{4}$ law of the structure function is justified.

Multi-Time Scaling for Phase Separation

The multi-time scaling for the phase separation is developed. Applying this scaling, the asymptotic form of the structure function is discussed. For the conservative system the structure function behaves as ∝ k 4 at small wave numbers k 's in the late stage of the phase separation. Using this asymptotic property a simplified structure function is given, which is found to agree well with experimental data.

Asymptotic Behavior of Structure Function at Small Wave Numbers in Late Stage of Phase Separation

The asymptotic behavior of the structure function at small wave numbers in systems undergoing phase separation is investigated. Brief reviews of the multi-time scaling and the k4-asymptotic behavior of the structure function for the Cahn-Hilliard equation are presented. Then the validity of the k 4-behavior for other systems is investigated. Numerical simulations for the spin exchange kinetic Ising model, for the Cahn-Hilliard model with the surface diffusivity only and for the simple liquid mixture are performed. It is found that the k4-behavior is consistent with these simulations.