The mode superposition method can be used to perform a time history analysis to obtain the response of structure due to a transient loading as a function of time. It requires the solution of Equation (6) for the frequencies and mode shapes.

where mode shapes can be shown to be orthogonal to the mass and stiffness matrices, as permit the equations of motion to be uncoupled.

The accelerations, velocities, and displacements in Equation (1) are transformed to a different coordinate system:

Substituting Equation (9) into Equation (1) and premultiplying by yields

where

―modal stiffness matrix;

―modal damping matrix;

―modal mass matrix;

―modal load vector.

Equation (1) can be decoupled by substituting Equations (7) and (8) into Equation (10).

where is the loading of the ith order mode, is the damping ratio for the ith mode and is the frequency of the ith order mode as following.

Equation (11) can be solved by a procedure for solving single-degree-of-freedom dynamic problems.

It should be mentioned that the higher mode shapes of the system are unimportant for a practical engineering structure or component. Neglecting the higher frequencies and mode shapes of the system generally does not introduce significant errors. Thus modal truncation is often considered to reduce the computational effort when the number of DOF is large.

Before the application of reduced basis method, a sample set of parameter domain is selected in a training space, which comprised of parameters spanning the parameter domain roughly.

The truncated eigenvectors corresponding to the parameters in the sample set are extracted to construct the reduced basis space

where m is the number of mode is retained in terms of the required accuracy.

It should be noted that the basis vectors are the solutions of the system equations at different parameters. They are perhaps nearly oriented in the same direction. Consequently, the resulted reduced system is very ill-posed especially for large, i.e. the condition number of the reduced stiffness matrix grows exponentially with. To guarantee the basis vectors’ linearly independence and make the reduced system well-posed, QR decomposition is used to generate a new basis which is orthogonal and able to retain all approximation properties of the original basis

The corresponding transform matrix is.

Then, the eigenvectors corresponding to a new parameter can be expressed as a linear combination of the basis vectors

The above equation also can be rewritten in a matrix form

To get the reduced system, the parameter-independent matrices are projected onto the reduced basis in terms of a Galerkin form.

From this parameter-decomposition expression, the reduced system can be easily obtained and explored in the whole parameter domain.

Obviously, the reduced eigenvalue problem can be solved more efficiently for each new parameter in test parameter-space.

The truncated eigenvectors can be regenerated approximately by

The approximation of eigenvalues can be demonstrated in terms of Rayleigh’s quotient.

The basis vectors selection is critical for the efficiency and accuracy of the reduced basis method. Too many or too few vectors selected should be avoided. The former results in computational inefficiency, while the latter in unacceptable error. To obtain an appropriate basis space, a greedy algorithm is suggested to select the vectors adaptively.

At first, the error in approximated eigenvalues is presented.

The maximum error in the training space is definite as

The performing procedure of greedy algorithm is summarized as follows.

Step 1. One parameter in the training space is selected as the start point; the associated truncated eigenvectors are extracted as the vectors of the reduced basis space.

Step 2. QR decomposition is applied to perform orthogonalization of basis vectors.

Step 3. The reduced generalized eigenvalue problem is solved in the training space to yield the approximated modes,.

Step 4. The maximum error is determined.

Step 5. The truncated eigenvectors corresponding to the maximum error will be selected as the next basis vectors and added to the reduced basis space. Then steps 2 to 4 are repeated. The greedy algorithm will terminate when the maximum error is less than a prescribed tolerance.

Direct integration provides a step-by-step numerical procedure to solve the equations of motion in Equation (1) directly without prior transformation of the equations to a different form. It can compute an approximate solution at discrete time intervals, where T is duration of the input motion or loading and is the time step. The widely used explicit methods are only conditionally stable such that some restrictions over the size of the selected time step. On the other hand implicit methods may be unconditionally stable, but the computational work and storage requirement per time step can be much greater than explicit methods because a solution of a coupled algebraic system is always involved. Generally, explicit methods are most preferred for wave propagation problems, while implicit methods are widely employed and advocated for structural dynamic problems.

Newmark method is considered as the example. It is a widely employed linear one-step implicit method with two basic assumptions

and is unconditionally stable under the following parameter limitation.

Given approximated values, , , for the displacement, velocity, and acceleration at time, the algorithmic values at time are the solution of the linear algebraic equations.

From the initial condition given by Equation (2), the initial acceleration given by

Just as the same in mode superposition, a sample set will be introduced from a training space comprised of a span of parameters and all time steps. The reduced basis space is defined as the span of finite element displacement response and doesn’t change with time for a new arbitrary parameter.

For the same reason mentioned in foregoing section, QR decomposition is applied to generate an orthogonal reduced basis space.

The transform matrix for projection can be written as:

The displacement response corresponding to new parameter and new time step can be approximated as the linear combination of the vectors in the reduced basis space.

It also can be expressed as a matrix form.

The approximated velocity and acceleration can be obtained by first order and second order derivatives of the approximated displacement response with respect to time, respectively.

The reduced Newmark formulation can be obtained by Galerkin projection of original space onto the reduced basis space.

The reduced stiffness, mass and damping matrices are respectively given by

where the reduced parameter-independent matrices are

The reduced load vector is

The initial condition corresponding to the reduced system is

Similarly, the greedy algorithm is adopted to select the vectors adaptively and subsequently obtain an appropriate reduced basis space.

At first, the projection error is defined in the training space as

where is the projection displacement,.

The maximum norm of the projection error is defined as

The perform procedure of greedy algorithm is summarized as follows.

Step 1. To span the training space, the displacement of the last time step is selected as the first basis vector, corresponding to one source in the training space.

Step 2. QR decomposition is applied to perform orthogonalization of basis vectors.

Step 3. The reduced Newmark’s formula is solved in the training space to yield the reduced basis displacements.

Step 4. The maximum norm of the projection error is determined.

Step 5. The displacement corresponding to the maximum norm of the projection error will be selected as the next basis vector and steps 2 to 4 are repeated. The greedy algorithm will terminate when the maximum norm of projection error is less than a prescribed tolerance.