Abstract
We present a method for dimensional reduction of the solution of the static equilibrium of a lattice-like network of point masses and harmonic springs. The dimensional reduction is achieved by energy-driven or goal-oriented hpadaptivity. A discrete version of the fully-automatic hp-adaptive algorithm is used to constrain the positions of individual masses to conform to a piecewise polynomial. The lattice is divided into elements that consist of adjacent masses and springs. Element refinement is done by reducing the number of masses in an element (h-refinement) or increasing the polynomial degree (p-refinement). The necessity of a refinement and its optimal type depends on a local error estimate. The elements chosen for further refinement are those that give maximum reduction in error per number of invested degrees of freedom. The method generalizes the well-known Quasicontinuum method for molecular simulations (Tadmor, Ortiz, and Phillips 1996). The presented numerical results confirm the exponential reduction in errors measured in the energy norm or in the quantity of interest. The method captures the variation in material constants by choosing large elements in regions of smooth variation of material data resulting in a significant dimensional reduction. The method fits into the “finite element” variational framework for boundary value problems. This enables utilization of an existing hp code for PDE problems with minimal changes.