*Forma,* Vol. 28 (No. 1), pp. 7–16, 2013

## Finite Elements for Geometrical Minimal Shape

Vinicius F. Arcaro^{1}*, Katalin K. Klinka^{2} and Dario A. Gasparini^{3}

^{1}University of Campinas, College of Civil Engineering, Av. Albert Einstein 951, Campinas, SP 13083, Brazil

^{2}Department of Structural Engineering, Budapest University of Technology and Economics, 2 Bertalan Lajos, Budapest, H-1111, Hungary

^{3}Civil Engineering Department, Case Western Reserve University, 2182 Adelbert Rd., Cleveland, OH 44106, U.S.A.

*E-mail address: vinicius.arcaro@arcaro.org

(Received November 2, 2012; Accepted August 23, 2013)

**Abstract. **
This text describes a novel mathematical model that unifies all geometrical minimal shape problems by defining
geometrical finite elements. Three types of elements are defined: line, triangle and tetrahedron. By associating
a volume for each element type, the elements can be used together in the discretization of a geometrical shape.
For each element type, its corresponding isovolumetric element is also defined. The geometrical minimal shape
problem is formulated as an equality constrained minimization problem. The importance of this approach is that
apparently distinct problems can be treated by a unified framework. The augmented Lagrangian method is used
to solve the associated unconstrained minimization problem. A quasi-Newton method is used, which avoids the
evaluation of the Hessian matrix. The source and executable computer codes of the algorithm are available for
download from the website of one of the authors.

**Keywords: **
Element, Finite, Form, Minimization, Nonlinear

[Full text] (PDF 213 KB)