INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS Int. J. Numer. Anal. Meth. Geomech., 2005; 29:1199–1207 Published online 27 June 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.454 Unified displacement boundary constraint formulation for discontinuous deformation analysis (DDA) David M. Doolin n,y,z Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720, U.S.A. SUMMARY Displacement boundary constraints in discontinuous deformation analysis (DDA) are applied using stiff penalty springs. A co-ordinate-free formulation for displacement boundary constraints is presented here for DDA, which unifies previous derivations for points of fixity, and for points constrained to induce or prohibit block motion in specified directions as a function of location or time. Examples for each type of constraint are used to illustrate the behaviour of the algorithm and provide a link with previous formulations for each case. The new, unified formulation has five benefits: (1) simple to express algorithmically; (2) easy to program and verify; (3) penalty values in different directions may be chosen to allow fixed points, lines, curves or planes; (4) formulation works for 2D and 3D; (5) displacement constraint may be a function of time or location or both. Feedback in the algorithm may induce internal resonance in homogeneously deformable discrete elements used in DDA, and resonance in block-to-block contact interactions. Consequently, high mass problems with insufficient damping may suffer from excessive ‘vibrational hammering’, inducing physically implausible behaviour such as elastic rebound. Copyright # 2005 John Wiley & Sons, Ltd. KEY WORDS: discontinuous deformation analysis; discrete element method; penalty method; feedback; vibrational hammering INTRODUCTION Discontinuous deformation analysis (DDA) is a discrete element method growing in popularity for geomechanical simulation. The DDA method is generally formulated using minimum potential energy [1–3] or by constructing a Lagrangian function and applying an appropriate variational principle [4]. Details of various potential energy and kinetic energy functionals have been widely reported, and are not necessary for discussion here. Boundary conditions in DDA are imposed using numerical springs, ‘fixing’ a point or points within a block to constrain the motion of the block to fixity, or along a certain direction, or according to a prespecified displacement. One end of the spring is attached to a location in a Received 3 December 2004 Revised 9 May 2005 Accepted 18 May 2005 Copyright # 2005 John Wiley & Sons, Ltd. y E-mail: doolin@ce.berkeley.edu z Assistant Research Engineer. n Correspondence to: David M. Doolin, 1301 S. 46th Street RFS 451, Richmond, CA 94804, U.S.A.