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The capability of the numerical discontinuous deformation analysis (DDA) method to perform site response analysis is tested. We begin with modeling one-dimensional shear wave propagation through a stack of horizontal layers and compare the obtained resonance frequency and amplification with results obtained with SHAKE. We use the algorithmic damping in DDA to condition the damping ratio in DDA by changing the time step size and use the same damping ratio in SHAKE to enable meaningful comparisons. We obtain a good agreement between DDA and SHAKE, even though DDA is used with first order approximation and with simply deformable blocks, proving that the original DDA formulation is suitable for modeling one-dimensional wave propagation problems. The ability of DDA to simulate wave propagation through structures is tested by comparing the resonance frequency obtained for a multidrum column when modeling it with DDA and testing it in the field using geophysical site response survey. When the numerical control parameters are properly selected, we obtain a reasonable agreement between DDA and the site response experiment in the field. We find that the choice of the contact spring stiffness, or the numerical penalty parameter, is directly related to the obtained resonance frequency in DDA. The best agreement with the field experiment is obtained with a relatively soft contact spring stiffness of k = (1/25)(EÂ L) where E and L are the Young's modulus and mean diameter of the drums in the tested column.

The discontinuous deformation analysis (DDA) is a discontinuum-based method, which employs a penalty method to represent the contact between blocks. The penalty method is easy to be implemented in the program, but the contact constraint is only approximately satisfied. Penetrations between contacting blocks are unavoidable even if the penalty value is very large. To improve the contact precision in the DDA, an augmented Lagrangian method is introduced, which can make use of advantages of both the Lagrangian multiplier method and the penalty method. This paper provides a detailed implementation of the augmented Lagrangian method in the DDA program and compares it with the standard DDA on the computational efficiency and contact precision.Compared with the Lagrangian multiplier method [3,4], penalty methods have an important advantage that no additional unknowns are introduced into the final system of equations and the global stiffness matrix is positive definite, which makes the implementation of penalty methods easy and straightforward [5][6][7]. Hence, penalty methods have been widely used in computer codes for the treatment of constraint conditions [8][9][10].In penalty methods, the contact force is assumed to be proportional to the penetration distance. Therefore, a penetration between blocks is inevitable if contact force exists. A drawback of this method is that the accuracy of the solution depends strongly on the choice of penalty values. In static problems, the penalty parameter should be, in principle, an arbitrarily large number to enforce the constraint condition. However, for a computer with a finite number of digits, it should not be so large that the governing equations become ill-conditioned. On the other hand, too small a penalty parameter may result in an unacceptable penetration of one body into the other, and the overall response is distorted.The augmented Lagrangian method (ALM) is an iterative method to obtain the exact solutions for contact forces with important advantages over the more traditional Lagrangian multiplier and penalty methods. It was initially proposed by Hestenes [11] and Powell [12] for solving non-linear programming problems with equality constraints. It was extended to treat convex differentiable optimization problems with inequality constraints (such as the frictionless contact problem) by Rockafellar [13]. Later, it was widely used in the contact problems of numerical methods [14][15][16][17][18][19][20][21][22][23][24][25].The essential concept behind the ALM is to use an augmented Lagrangian multiplier that can be iteratively calculated by adding the penalty force onto the Lagrangian multiplier obtained in last iteration. In the ALM, the Lagrangian multiplier is not unknown any more as that in the Lagrangian multiplier method. Hence, the simplicity of the penalty method is retained, and its disadvantages are minimized. The precision of solutions depends on residual forces, that is, the penalty forces produced during the current iteration. The residual force should be as ...