Modeling of Nanostructure Vibrations using a Finite Elements MethodElements Method

The calculations in the case of the one dimensional confinement were treated analytical and numerical. One-dimensional analytical estimates were used to derive the eigenfre-quencies, and numerical modeling was used for the estimation of the amplitudes in the Fourier spectra. In the case of higher order confinement, the calculations are far more complex to solve, if not impossible. Therefore a 3D modeling system using the finite element method was applied.

This section gives a short overview over the technique of finite element methods (FEM), or FEM simulations, illustrating where these methods are useful to implement, and explaining the advantages and disadvantages of these methods. In order to understand the underlying concept, the static approach of FEM is illustrated on the example of a one dimensional bar. The section closes with the description of the approach of solving dynamical problems, as this approach was used in this chapter. The argumentation and description in this section follows Reference [146].

FEM simulations are commonly used in engineering or science when complex geometric structures, properties and/or boundary conditions prohibit to solve the desired question analytically within a reasonable amount of time, if possible at all. The basic idea of this technique is to divide the complex problem into smaller, more simple problems. The result is an approximation of the real problem, not an exact solution. By dividing the original problem/structure into a finite number of subdomains, the approximation is also finite. This is an aspect to keep in mind when discussing FEM results. Also artifacts and influences of the discretization can influence the result, or give solutions which are not realistic at all.

5 Experimental Results - Higher Order Confinement

FEM can can be easily applied to complex, irregular shaped models, which are composed of several different materials. It has found versatile application to steady-state, time-dependent and eigenvalue problems, as well in linear as in nonlinear problems. Modeling can include for example heat transfer, fluids or complex loads. FEM provide an “easy”

combination of different physical models, i.e. electro-magnetics, fluids, mechanics, etc.

There are also several disadvantages to FEM. The main disadvantage is that a specific result is for a specific problem only. No general, closed-form solutions, as obtained from analytical models, can be derived from a FEM solution. This means that a generalization of the results obtained by FEM simulations is difficult, since FEM is an approximation of a mathematical model only. Also numerical problems can result in wrong predictions from these models. The three main causes of faulty results of FEM can be summed up by:

discretization poor choice of elements shape or size, boundary shape, constraints formulation poor choice of elements, i.e. element choice based on the assumption of

linear behavior applied in a nonlinear problem

numerical number of significant digits (truncation), round on/off error accumula-tion

Overall, in order to obtain reliable results, the use of FEM requires well-thought assump-tions for the preparation of a model, as well as experience in interpreting the results.

Implementation Procedure of FEM Simulations

The implementation procedure of FEM simulations can be described in five steps, as stated below:

1. Discretization: divide the model into the desired number of sub-domains. The accuracy increases with the number of elements, but so does the necessary com-putational power. Symmetry considerations can greatly reduce the computation time.

2. Define the stiffness matrix for each sub-domain

3. Assembly: assemble the sub-domains to form the overall structure, define the boundary conditions, etc.

4. Solution of the problem, i.e. static or dynamical solution solver 5. Solution of all other parameters, e.g. stress, strain, . . .

The solver in step 4 is chosen according to the desired problem. In more general cases, multiple solvers can be combined. These five steps are illustrated in the next para-graph.

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5.1 Introduction: Experiments on Single Nanostructures

Static Approach of FEM Simulation

The following descriptions are adopted from Reference [146]. The static approach of an FEM simulation is explained in the following with the help of the example of the simulation of the effects of static load onto an object, i.e. a one-dimensional bar. The problem is illustrated in Figure 5.1. A typical statical question could be the effect of a point load at the end of the bar, while the bar is fixed at the wall.

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Figure 5.1: FEM-nodes and elements illustrated on the two-dimensional example of a bar with a fixed and an open end. The bar is divided into five elements (1-5). At each boundary between two elements sits a node (I-VI). At these nodes the boundary conditions of displacement and load are applied, connecting the neighboring elements.

Inside each element a linear distribution of dis-placement and load is assumed.

Steps 1-3: The bar is modeled with a dis-crete number of elements (1-5). In one di-mension, each element connects two nodes (I-VI). The nodes describe the interaction between the elements, in this case using Q for the displacement and F for the load.

Within one element a linear interpolation is applied between the two nodes, therefor smaller elements result in a higher simu-lation accuracy. Mechanical calcusimu-lations between elements follow the same rules (e.g. boundary conditions, continuity at boundaries, etc.) as described in Sec-tion 2.2: Numerical Modeling. The point load would be applied at the node VI.

Step 4: The iteration process calculates the effect of the point load for each ele-ment and at each node. The solution is de-rived via the Potential Energy Approach, which states that minimizing the poten-tial energy yields the possible solutions.

This approach follows from the Minimum Potential Energy Theorem: “Of all possi-ble displacements that satisfy the bound-ary conditions of a structural problem, those corresponding to equilibrium

config-urations make the potential energy assume a minimum value.”. From this theorem, it follows that under considerations of the boundary conditions, minimizing the potential energy yields the possible solutions of the problem [146].

Dynamical Solutions

Dynamical solutions, e.g. eigenfrequencies of a system, can be obtained by solving the corresponding eigenvalue problem. This is done in the FEM simulations in this thesis in order to obtain the eigenfrequencies of the nanostructures.

5 Experimental Results - Higher Order Confinement

Taking the potential energy and the Hamilton principle, it is possible to solve the gen-eralized eigenvalue problem. The method applied is the inverse iteration scheme. This scheme uses a probe vector, e.g. for the displacement in z direction u^z₀ = Asin(ωz), where A is the amplitude, to find the lowest eigenvalue for this probe vector. The problem then writes as

K~u=λM ~u, (5.1)

whereK denotes the stiffness matrix, M the mass matrix and~uthe eigenvector for the test frequency with the eigenvalue λ=ω² =√

2πf.

The inverse iteration scheme follows numerically the procedures described below [146].

Step 1: Estimate an initial trial vector: ~u₀. Iteration index: k=0.

Step 2: Set k=k+ 1.

Step 3: Determine right side of~v^k−1 =M ~u^k−1. Step 4: Solve equation: K~uˆ^k =~v^k−1.

Step 5: Denote~ˆv^k=M~uˆ^k.

Step 6: Estimate eigenvalue: λ^k= ^~ˆvkT_ˆ ^~vˆk−1

~u^kT~ˆv^k . Step 7: Normalize eigenvector: ~ˆv^k = √^ˆ~vk

~ˆ u^kT~ˆv^k. Step 8: Check for tolerance:

λ^k−λ^k−1 λ^k

≤tolerance.

This algorithm converges to the lowest eigenvalue of the probe vector, given that the probe vector is not one of the eigenvectors. Other eigenvalues can be evaluated by shifting the probe vector. This procedure is used in Comsol to find the mechanical eigenfrequencies in the simulations of the beam and disk nanostructures investigated in this thesis.

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