Development of displacement hp-version space-time finite elements with triangular elements

A bilinear formulation of space-time dynamics is presented that numerical studies can be performed for h(element size) or p-version (order of polynomial) elements of any shape. The commitment to a weak formulation allows appropriate solution discontinuities to develop naturally a t element boundaries. To predict the steep gradients of stress in wave-like motions, triangular elements are used to cut across space and time such that discontinuities in force or momentum always located a t the element boundary. The shape functions for triangular elements are derived in terms of a set of orthogonal functions, and the assembly procedure for hp-version space-time finite elements is also discussed. I n t r o d u c t i o n Since the equivalence between boundary value problem of partial differential equations and the problems of calculus of variations was observed, both theoretical proofs and pratical applications developed rapidly. The idea of utilizing this equivalence for numerical solutions soon attracted engineers and physicists. One of the most widely used and successful numerical methods based on minimum principles is the finite element method, which has been frequently used for spatial problems. Direct time-marching methods were standard procedures for the solution of initial-value problems in the time domain. In the field of structural dynamics of continuous systems, space and time are usually separated due t o the fact that space is a boundary-value problem and time is an initial-value problem. Thus, solution methods solve the spatial solution a t any time step before proceeding to the next step. Computational difficulties for these conventional strategies can arise when structures are subjected to impulsive loadings, because a large number of degrees of freedom are required to model such structures a t points where steep gradients in velocity and displacement field 'Graduate Research Assistant, Member A H S Copyright @ 1992 by Lin-Jun Hou. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. occurred. Also, the flexibility of refining the mesh in the space-time domain diminished as the discretization is performed first in space and then independently in time. A major alternative, finite difference in space and time, could solve these problems; but it is computationally inefficient for use in design-oriented analysis. Not until the last decade, did people apply Hamilton's Law of varying action to develop solution strategies of temporal problems; and in which the "action" in the time domain can be treated as a counterpart of energy in the space domain. Time finite elements, which researchers have been studying as an alternative method of solution, are more compatible with finite difference and conventional methodologies. There are several potential advantages to such approaches : 1). they could be applied to the energy (or action) rather than to the differential equations, thus saving the cost associated with derivation of these equations, 2). finite elements in time could be tuned to give minimum error a t points for which the greatest accuracy is desired. Thus, time finite elements offer a unified solution strategy for the "space-time" domain which could give synergistic improvements in computational efficiency. Some works on space-time finite elements either separated the spatial and time variables thus lead to a time integration scheme or formulated the equations in a weighted residual sense in time thus implicitly structured the space-time mesh 3. Several other papers 4-s have shown some successful results for finite elements in the time or space-time domain. In this paper, a weaker displacement hp-version space-time finite element is derived from Hamilton's weak principle 13 (HWP) and prcsented for sin~ultaneous space-time solutions. A special type of discretization of space-time is i~lvolved, in which both space and time dependency are removed by expansion of the unknowns in terms of polynomials. Although there are several choices for the displacement methods, we presently prefer the one which only displacements are strongly enforced, and both the trial and test functions are chosen from the class CO. Since no matter what conditions are imposed on the space-time boundaries, the equations and unknowns match for any h p combination without special considerations. F o r m u l a t i o n o f T h e o r y The theoretical basis stems from the Hamilton's weak principle, from which we s tar t with the methodology based on a variational statement of elasto-dynamics by Peters l o : "the variation of the action plus the virtual action over the time interval plus the net virtual action across the time boundaries must sum to zero" or where s denotes the "surface" of the space-time domain, L: the Lagrangian density and 6W, the virtual work per unit length. Along space boundaries, the g is the force and ds = dt which makes the trailing term in Eq. (1) the time integral of virtual work (virtual action). Along time boundaries, the quantity g is the momentum per unit length a t the beginning or end of the time frame and ds = dx. When Eq. (1) is taken over an element which crosses space and time, the trailing terms become the force-momentum balance acrosses the moving boundary. This is especially useful for moving constraints and deployment problems. Several possibilities of the variational formulations are valid depend on the degrees of integration by parts, and on various combinations of weaklyor stronglyenforced conditions. Further, enforcing the Lax-Milgram Lemma " will eliminate all the combinations that involve derivatives of unknowns explicitly in the boundary terms. This leaves us with three displacement formulations. Of these, only one displacement formulation (bilinear) can be used with triangular elements, which have the advantage that each hierarchical level has consistent powers of x and t. Disp lacement F o r m u l a t i o n s One typical example is the longitudinal vibration of a uniform rod surrounded with elastic foundation. The basic equations are : t Differential Equation : F p + f ku = 0 Constitutive Laws : p = m i , f = put dx dt Boundary Momentum Flux : g = p + f ds ds dx t dt Strong Version of g : h = m i + pu ds ds Boundary Conditions : u r ii on I?, g E ij on A where F is the external applied force, and r*, A* are complementary pieces of boundary. For well defined problem, I? and .I* (or r* and A) span same space of functions. The ( ) quantities represent the values on the boundary. The displacement formulation obtained from the variational statement of elasto-dynamics (Eq.(l)) can be written as follows F6u + m i 6 i pu16u' ku6u dzdt I + i Aauds + i ( u ii)Shds = 0 (2)