^{1}

^{*}

^{2}

The reduced basis methods (RBM) have been demonstrated as a promising numerical technique for statics problems and are extended to structural dynamic problems in this paper. Direct step-by-step integration and mode superposition are the most widely used methods in the field of the finite element analysis of structural dynamic response and solid mechanics. Herein these two methods are both transformed into reduced forms according to the proposed reduced basis methods. To generate a reduced surrogate model with small size, a greedy algorithm is suggested to construct sample set and reduced basis space adaptively in a prescribed training parameter space. For mode superposition method, the reduced basis space comprises the truncated eigenvectors from generalized eigenvalue problem associated with selected sample parameters. The reduced generalized eigenvalue problem is obtained by the projection of original generalized eigenvalue problem onto the reduced basis space. In the situation of direct integration, the solutions of the original increment formulation corresponding to the sample set are extracted to construct the reduced basis space. The reduced increment formulation is formed by the same method as mode superposition method. Numerical example is given in Section 5 to validate the efficiency of the presented reduced basis methods for structural dynamic problems.

Nowadays structural dynamic problems are usually solved by the finite element technique. Solution of displacement responses of all the nodes requires great effort. The scale and complexity of dynamics problems of practical engineering structure are ever increasing such that it requests more memory and computing time than before. Despite of the continuing advances in computer speeds and hardware capabilities, the dimension for numerical simulation is too large to provide real-time response in the design, optimization, control and characterization of engineering components or systems. Thus there are many motivations to develop methods that can not only reduce significantly the problem size and computational cost but also retain the accuracy of the solution and the physics of the structures.

Model order reduction techniques [

However, order reduction has long been focused on control problems [

Different from the traditional reduction methods, the reduced basis method (RBM) [

In this paper we adopt the reduced basis method to perform the dynamic analysis of structures based on mode superposition method and direct integration method, respectively. A greedy algorithm is suggested to perform the adaptively selection of reduced basis vectors. Numerical example of a simplified one-dimensional seismic model is presented to demonstrate the feasible application of reduced basis method in structural dynamic problems. The error of the reduced system is evaluated numerically.

In structural dynamic analysis, the equations of motion are generally written as a set of linear second-order differential equations. The matrix form of these equations may be expressed by:

where

equivalent force vector acting on the nodes; the total mass matrix

and

In the following analysis, the stiffness and mass matrices are assumed as parameter-decomposition forms

In Equation (3) and Equation (4),

The damping matrix is considered to be proportional.

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

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

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

where

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

where

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

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

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

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

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

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

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

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

A simplified one-dimensional seismic model [

The earthquake source

denote the spatial distribution and the temporal characteristics of earthquake source respectively, are showed in

To obtain parameter-decomposition forms of stiffness and mass matrices, namely, to express the stiffness and

mass matrices as the combination form of product of system parameter function and the matrix independent of

system parameters. The original x-domain is decomposed into the left zone

A piecewise affine mapping from the standard y-domain to the original x-domain is given in

where

As the reduced structural dynamic analysis performed by using mode superposition, the 12^{th} truncation of mode is considered. It can be found from

The resulted reduced eigenvalue problem is 60 in dimensional, while the reduced Newmark formulation is 85 in dimensional for a prescribed error tolerance

Method | Newmark | Reduced Newmark | Mode superposition | Reduced mode superposition |
---|---|---|---|---|

CPU Time (s) | 6.0156 | 1.3281 | 3.6406 | 2.0625 |

space. Despite the original mode superposition more effectively executed than the original Newmark method, the reduced form of the former costs more CPU time than the reduced form of the later. It can be concluded that the dynamic analysis have been performed much more effectively by either reduced mode superposition or reduced Newmark method.

It should be point out that the dimensional of the reduced system is determined by the reduced basis space and independent of the original system. For larger dynamic system, the efficiency of the reduced basis methods can be further enhanced.

Two kinds of reduced basis methods for dynamic problems are proposed in this paper. In the numerical example, the direct integration for the dynamic analysis is not numerically efficient as compared with the mode superposition method using eigenvectors due to the linear property of the seismic problem. However, it proves that the reduced basis method is available for structural dynamic analysis based on either mode superposition or direct integration. Though the undamped case studied, the reduced basis method can be applied to damped structures without any more effort. Furthermore, although the reduced Newmark method is only considered here, the reduced basis method can be very easily extended to other direct integration techniques.

This project is supported by National Natural Science Foundation of China (Grant No. 51305045), and by China Postdoctoral Science Foundation (No. 2014M562099).

YonghuiHuang,YiHuang, (2015) Adaptive Reduced Basis Methods Applied to Structural Dynamic Analysis. American Journal of Computational Mathematics,05,317-328. doi: 10.4236/ajcm.2015.53029