The extended unsymmetric frontal solution for multiple-point constraints

Pedro Miguel de Almeida Areias (Department of Physics. Universidade de Évora, Évora, Portugal)
Timon Rabczuk (Facultat Bauingenieurwesen, Bauhaus-Universitat, Weimar, Germany)
Joaquim Infante Barbosa (Department of Mechanical Engineering, Instituto Superior de Engenharia de Lisboa, Lisboa, Portugal)

Engineering Computations

ISSN: 0264-4401

Publication date: 30 September 2014



The purpose of this paper is to discuss the linear solution of equality constrained problems by using the Frontal solution method without explicit assembling.


Re-written frontal solution method with a priori pivot and front sequence. OpenMP parallelization, nearly linear (in elimination and substitution) up to 40 threads. Constraints enforced at the local assembling stage.


When compared with both standard sparse solvers and classical frontal implementations, memory requirements and code size are significantly reduced.

Research limitations/implications

Large, non-linear problems with constraints typically make use of the Newton method with Lagrange multipliers. In the context of the solution of problems with large number of constraints, the matrix transformation methods (MTM) are often more cost-effective. The paper presents a complete solution, with topological ordering, for this problem.

Practical implications

A complete software package in Fortran 2003 is described. Examples of clique-based problems are shown with large systems solved in core.

Social implications

More realistic non-linear problems can be solved with this Frontal code at the core of the Newton method.


Use of topological ordering of constraints. A-priori pivot and front sequences. No need for symbolic assembling. Constraints treated at the core of the Frontal solver. Use of OpenMP in the main Frontal loop, now quantified. Availability of Software.



The first author gratefully acknowledges the generous help from Professor Iain Duff in obtaining technical documentation of the frontal solution method. He is also grateful to Professor R. Owen for the three-month stay in UCS, Wales, UK in the year 1999, where interest in the topic emerged. The authors also gratefully acknowledges the availability of several HSL codes for testing (cf. HSL, 2011). The authors gratefully acknowledge financing from the “Fundação para a Ciência e a Tecnologia” under the Project PTDC/EME-PME/108751 and the Program COMPETE FCOMP-01-0124-FEDER-010267.


Miguel de Almeida Areias, P., Rabczuk, T. and Infante Barbosa, J. (2014), "The extended unsymmetric frontal solution for multiple-point constraints", Engineering Computations, Vol. 31 No. 7, pp. 1582-1607.

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