Modes used for ROM can either be determined from the results of the test load application or based on their modal stiffness at the initial position.

Case 1: Test Load is Available (TMOD option on RMMSELECT command)

The test load drives the structure to a typical deformation state, which is representative for most load situations in the Use Pass. The mode contribution factors a_i are determined from

(14–145)

where:

Mode contribution factors a_i are necessary to determine what modes are used and their amplitude range. Note that only those modes are considered in Equation 14–145, which actually act in the operating direction (specified on the RMANL command). Criterion is that the maximum of the modal displacement in operating direction is at least 50% of the maximum displacement amplitude. The solution vector a_i indicates how much each mode contributes to the deflection state. A specified number of modes (Nmode of the RMMSELECT command) are considered unless the mode contribution factors are less than 0.1%.

Equation 14–145 solved by the least squares method and the results are scaled in such a way that the sum of all m mode contribution factors a_i is equal to one. Modes with highest a_i are suggested as basis functions.

Usually the first two modes are declared as dominant. The second mode is not dominant if either its eigenfrequency is higher than five times the frequency of the first mode, or its mode contribution factor is smaller than 10%.

The operating range of each mode is proportional to their mode contribution factors taking into account the total deflection range (Dmax and Dmin input on the RMMSELECT command). Modal amplitudes smaller than 2.5% of Dmax are increased automatically in order to prevent numerical round-off errors.

Case 2: Test Load is not Available (NMOD option on RMMSELECT command)

The first Nmode eigenmodes in the operating direction are chosen as basis functions. Likewise, a considered mode must have a modal displacement maximum in operating direction of 50% with respect to the modal amplitude.

The minimum and maximum operating range of each mode is determined by:

(14–146)

where:

Up to 5 element loads such as acting gravity, external acceleration or a pressure difference may be specified in the Generation Pass and then scaled and superimposed in the Use Pass. In the same way as mode contribution factors a_i are determined for the test load, the mode contribution factors for each element load case are determined by a least squares fit:

(14–147)

where:

= displacements at the neutral plane nodes obtained from the results of the element load j (ELOAD option on RMNDISP command).

Here index k represents the number of modes, which have been selected for the ROM. The coefficients are used to calculate modal element forces (see Element Matrices and Load Vectors).

In a general case, the energy functions depend on all basis functions. In the case of m modes and k data points in each mode direction one would need k^m sample points.

A large number of examples have shown that lower eigenmodes affect all modes strongly whereby interactions among higher eigenmodes are negligible. An explanation for this statement is that lower modes are characterized by large amplitudes, which substantially change the operating point of the system. On the other hand, the amplitudes of higher modes are reasonably small, and they do not influence the operating point.

Taking advantage of those properties is a core step in reducing the computational effort. After the mode selection procedure, the lowest modes are classified into dominant and relevant. For the dominant modes, the number of data points in the mode direction defaults to 11 and 5 respectively for the first and second dominant modes respectively. The default number of steps for relevant modes is 3. Larger (than the default above) number of steps can be specified on the RMMRANGE command.

A very important advantage of the ROM approach is that all finite element data can be extracted from a series of single domain runs. First, the structure is displaced to the linear combinations of eigenmodes by imposing displacement constrains to the neutral plane nodes. Then a static analysis is performed at each data point to determine the strain energy.

Both the sample point generation and the energy computation are controlled by the command RMSMPLE.

The objective of function fit is to represent the acquired FE data in a closed form so that the ROM FE element matrices (ROM144 - Reduced Order Electrostatic-Structural) are easily derived from the analytical representations of energy functions.

The ROM tool uses polynomials to fit the energy functions. Polynomials are very convenient since they can capture smooth functions with high accuracy, can be described by a few parameters and allow a simple computation of their local derivatives. Moreover, strain energy functions are inherent polynomials. In the case of linear systems, the strain energy can be exactly described by a polynomial of order two since the stiffness is constant. Stress-stiffened problems are captured by polynomials of order four.

The energy function fit procedure (RMRGENERATE command) calculates n_c coefficients that fit a polynomial to the n values of strain energy:

(14–148)

where:

Note that the number of FE data (W_SENE) points n for a mode must be larger than the polynomial order P for the corresponding mode (input on RMPORDER command). Equation 14–148 is solved by means of a least squares method since the number of FE data points n is usually much larger than the number polynomial coefficients n_c.

The ROM tool uses four polynomial types (input on RMROPTIONS command):

Lagrange and Pascal coefficient terms that form matrix [A] in Equation 14–148 are shown in Figure 14.8: Set for Lagrange and Pascal Polynomials.

Figure 14.8:  Set for Lagrange and Pascal Polynomials

Polynomials for Order 3 for Three Modes (1-x, 2-y, 3-z)

Reduced Lagrange and Reduced Pascal polynomial types allow a further reduction of K_POLY by considering only coefficients located on the surface of the brick and pyramid respectively.

The ROM method is applicable to electrostatic-structural systems.

The constitutive equations for a coupled electrostatic-structural system in modal coordinates are:

(14–149)

for the modal amplitudes and

(14–150)

where:

The electrostatic co-energy is given by:

(14–151)

where:

The capacitances C_ij, and the electrostatic co-energy respectively, are functions of the modal coordinates q_i. As the strain energy W_SENE for the structural domain, the lumped capacitances are calculated for each k data points in each mode direction, and then fitted to polynomials. Following each structural analysis to determine the strain energy W_SENE, (n-1) linear simulations are performed in the deformed electrostatic domain, where n is the number of conductors, to calculate the lumped capacitances. The capacitance data fit is similar to the strain energy fit described above (Function Fit Methods for Strain Energy). It is sometimes necessary to fit the inverted capacitance function (using the Invert option on the RMROPTIONS command).