Multiple-scale analysis Wikipedia

For simple fluids, this will result in the same Navier-Stokes equation we derived earlier, now with a formula for \(\mu\) in terms of the output from the microscopic model. But for complex fluids, this would result in rather different kinds of models. Analytical asymptotic approximations for large amplitude nonlinear free vibration of a dielectric elastomer balloon have been studied by Tang et al. They presented Newton–harmonic balance method to investigate effects of initial stretch on vibrations of a dielectric elastomer balloon (Tang et al., 2017).

What is the multiple scales method

The idea is to decompose the whole computational domain into several overlapping or non-overlapping subdomains and to obtain the numerical solution over the whole domain by iterating over the solutions on these subdomains. The domain decomposition method is not limited to multiscale problems, but it can be used for multiscale problems. When studying chemical reactions involving large molecules, it often happens that the active areas of the molecules involved in the reaction are rather small. The rest of the molecules just serves to provide the environment for the reaction. In this case, it is natural to only treat the reaction zone quantum mechanically, and treat the rest using classical description. Such a methodology is called the QM-MM (quantum mechanics-molecular mechanics) method .

Quantum mechanics – molecular mechanics (QM-MM) methods

Bifurcation and periodic solutions of a viscoelastic dielectric elastomer balloon have been investigated by Liu and Zhou. They employed the Euler-Lagrange method to obtain the governing equation and solved the problem using the method of shooting and arc-length continuation . While in the studies above, numerical methods most commonly have been applied to investigate the vibration of DEBs, in the fewer studies, have been focused on analytical approaches. However, the reliable and accurate analytical solutions are necessary for some engineering conditions. Dielectric elastomers are widely used in several applications such as artificial muscles, sensors or energy harvesters.

Homogenization methods can be applied to many other problems of this type, in which a heterogeneous behavior is approximated at the large scale by a slowly varying or homogeneous behavior. The structure of such an algorithm follows that of the traditional multi-grid method. In a two-level setup, at any macro time step or macro iteration step, the procedure is as follows. Another important ingredient is how one terminates the quantum mechanical region, in particular, the covalent bonds.

Their hyperelastic nature complicates their design because of the obtained rich dynamics. The electrical field applied in the material, couples the dynamics of these devices to the applied voltage through compliant electrodes. In this work, a circular DE membrane under mechanical and electrical excitations is analyzed in order to seek internal resonances capable of increasing its performance for future applications. The hyperelastic properties of the material and the Maxwell stresses are taken into account to derive the nonlinear governing equation.

It should be noted that HMM represents a compromise between accuracy and feasibility, since it requires a preconceived form of the macroscale model to begin with. To see why this is necessary, just note that even for the situation when we do know the macroscale model in complete detail, selecting the right algorithm to solve the macroscale model is still often a non-trivial matter. Therefore trying to capture the macroscale behavior without any knowledge about the macroscale model is quite difficult. Of course, the usefulness of HMM depends on how much prior knowledge one has about the macroscale model. In particular, guessing the wrong form of the macroscale model is likely going to lead to wrong results using HMM. The complex scaling method is one of the most powerful methods of describing the resonances with complex energy eigenstates based on non-Hermitian quantum mechanics.

The outcome of the numerical results indicates that both the chaos and quasiperiodicity arise in the micro reasoner. Besides, the Lame’s modulus could harness the occurrence of the chaotic motion in the system. There are many materials that the linear elasticity theory cannot predict their mechanical behavior against applied loads. This research proposes a comprehensive theoretical method to obtain the mechanical response of hyperelastic models such as polymers and rubbers. They are vital in the design phase of complicated engineering structures like engine mounts and structural bearings in aerospace and automotive industries. The presented theory is implemented in detail and has no limitations in analyzing geometrically and physically nonlinear materials.

The new, previously unreported vibration and jump phenomena demonstrated in this paper may be leveraged for improved performance in applications like resonators, energy harvesters, actuators, and sensors. Nevertheless, the harmonic balance method needs to take into account the higher harmonic terms in the process of solving the problem which leads to difficult algebraic manipulation (Nayfeh and Mook, 2008; Nayfeh, 2011). The multiple scales method is one of the strongest analytical methods that belongs to the perturbation theory. The MMS constructs uniformly valid approximations to the solutions of the nonlinear problems. This chapter deals only with the two-time scaling because the other methods are, due to many but trivial calculations, rather complicated.

Multiple Scales Method

The hope is that by using such a multi-scale (and multi-physics) approach, one might be able to strike a balance between accuracy and feasibility . Roughly speaking, one might regard HMM as an example of the top-down approach and the equation-free as an example of the bottom-up approach. In HMM, the starting point is the macroscale model, the microscale model is used to supplement the missing data in the macroscale model.

Revealing the complex magnetization reversal mechanism with … – Science Daily

Revealing the complex magnetization reversal mechanism with ….

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Pneumatically connected DE actuator is one of the prevalently used configurations, wherein an electrically driven DE membrane is connected to another passive DE membrane via an intervening air column. This paper outlines the development of a theoretical model, well-supported by experimental observations, for analyzing the dynamic electromechanical response of PCDE actuators. An idea of pseudo air-spring, responsible for the transfer of deformation from the active to passive sides of the actuator, is introduced and the analytical expressions are developed for the coupling function.

In the equation-free approach, particularly patch dynamics or the gap-tooth scheme, the starting point is the microscale model. Various tricks are then used to entice the microscale simulations on small domains to behave like a full simulation on the whole domain. In the heterogeneous multiscale method , one starts with a preconceived form of the macroscale model with possible missing components, and then estimate the needed data from the microscale model. Quasicontinuum method (Tadmor, Ortiz and Phillips, 1996; Knap and Ortiz, 2001) is a finite element type of method for analyzing the mechanical behavior of crystalline solids based on atomistic models. A triangulation of the physical domain is formed using a subset of the atoms, the representative atoms (or rep-atoms). In regions where the deformation gradient is large, more atoms are selected.

Traditional approaches to modeling

Zhu et al. have presented the harmonic balance method to analytically study the nonlinear oscillation of a dielectric elastomer balloon (Zhu et al., 2010). Tang et al., have analyzed dynamic response and stability analysis with Newton harmonic balance method for nonlinear oscillating dielectric Elastomer balloons. They qualitatively and quantitatively investigated the effects of different factors such as pressure, voltage, and initial stretch ratio on the dynamic behavior of an elastomeric balloon https://xcritical.com/ (Tang et al., 2018). The need for multiscale modeling comes usually from the fact that the available macroscale models are not accurate enough, and the microscale models are not efficient enough and/or offer too much information. By combining both viewpoints, one hopes to arrive at a reasonable compromise between accuracy and efficiency. Dielectric elastomers represent a class of electroactive polymers that are capable of undergoing large reversible deformations when driven electrically.

  • Here the macroscale variable \(U\) may enter the system via some constraints, \(d\) is the data needed in order to set up the microscale model.
  • Because extremely large stretches are possible without snap-through instability, resonances in the frequency response transition from softening nonlinearity at moderate stretches to hardening nonlinearity at large stretches.
  • In the equation-free approach, particularly patch dynamics or the gap-tooth scheme, the starting point is the microscale model.
  • The idea is to divide the domain of interest into inner and outer regions, and introduce inner variables in the inner region, with the goal that in the new variables, the solutions have \(\mathcal\) gradients.
  • In the heterogeneous multiscale method , one starts with a preconceived form of the macroscale model with possible missing components, and then estimate the needed data from the microscale model.
  • To ensure that there are no secular terms in \(Y_1\ ,\) the resonant terms on the right hand side of are forced to be zero, i.e.
  • The dynamic stretches therefore can be extremely large, and are limited only by the locking stretch of the elastomer.

Both pressurized and depressurized states are considered in the analysis. The proposed model predicts the transfer efficiency of about 80% between the active and the passive membranes. The predictive capability of the theoretical model is established through comparisons with the experimental observations.

In this section, the MMS is applied to the two cases of the equation of motion to obtain the reliable analytical approximate solution for them. Based on whatever knowledge that is available on the possible form of the macroscale model, one selects a suitable macroscale solver. For example, if we are dealing with a variational problem, we may use a finite element method as the macroscale solver.

Nonlinear dynamics and bifurcation behavior of a sandwiched micro-beam resonator consist of hyper-elastic dielectric film

The idea is to divide the domain of interest into inner and outer regions, and introduce inner variables in the inner region, with the goal that in the new variables, the solutions have \(\mathcal\) gradients. Partly for this reason, the same approach has been followed in modeling complex fluids, such as polymeric fluids. In order to model the complex rheological properties of polymer fluids, one is forced to make more complicated constitutive assumptions with more and more parameters. For polymer fluids we are often interested in understanding how the conformation of the polymer interacts with the flow. This kind of information is missing in the kind of empirical approach described above.

What is the multiple scales method

When the system varies on a macroscopic scale, these conserved densities also vary, and their dynamics is described by a set of hydrodynamic equations . In this case, locally, the microscopic state of the system is close to some local equilibrium states parametrized by the local values of the conserved densities. Here the macroscale variable \(U\) may enter the system via some constraints, \(d\) is the data needed in order to set up the microscale model. For example, if the microscale model is the NVT ensemble of molecular dynamics, \(d\) might be the temperature. Macroscale models require constitutive relations which are almost always obtained empirically, by guessing.

Internal resonance and nonlinear dynamics of a dielectric elastomer circular membrane

These effects could be insignificant on short time scales but become important on long time scales. Classical perturbation methods generally break down because of resonances that lead to what are called secular terms. Matched asymptotics is a way of extracting the local structure of singularities or sharp transition layers in solutions of differential equations.

Classically this is a way of solving the system of algebraic equations that arise from discretizing differential equations by simultaneously using different levels of grids. In this way, one can more efficiently eliminate the errors on different scales using different grids. In particular, it is typically much more efficient to eliminate large scale component of the errors using coarse grids. The first is that the implementation of CPMD is based on an extended Lagrangianframework by considering the wavefunctions for electrons in the same setting as the positions of the nuclei.

Acta Mech. Solida Sin.

In sequential multiscale modeling, one has a macroscale model in which some details of the constitutive relations are precomputed using microscale models. For example, if the macroscale model is the gas dynamics equation, then an equation of state is needed. When performing molecular dynamics simulation using empirical potentials, one assumes a functional form of the empirical potential, the parameters multi-scale analysis in the potential are precomputed using quantum mechanics. This work demonstrates novel nonlinear vibration behavior in circular dielectric elastomer membranes that are excited by low nominal voltages with moderate-to-large-amplitude sinusoidal fluctuations. Low nominal voltages are those where the membrane has only one equilibrium stretch, and snap-though instability is not possible.

Alphanumerical scales

The length of the line on the linear scale is equal to the distance represented on the earth multiplied by the map or chart’s scale. This website is using a security service to protect itself from online attacks. There are several actions that could trigger this block including submitting a certain word or phrase, a SQL command or malformed data. This is a general strategy of decomposing functions or more generally signals into components at different scales.

Dynamic electromechanical instability of a dielectric elastomer balloon

This is a strategy for choosing the numerical grid or mesh adaptively based on what is known about the current approximation to the numerical solution. Usually one finds a local error indicator from the available numerical solution based on which one modifies the mesh in order to find a better numerical solution. This is a way of summing up long range interaction potentials for a large set of particles. The contribution to the interaction potential is decomposed into components with different scales and these different contributions are evaluated at different levels in a hierarchy of grids. Even though the polymer model is still empirical, such an approach usually provides a better physical picture than models based on empirical constitutive laws. The first scheme to address this problem is what Van Dyke refers to as the method of strained coordinates.