I want to thank my mentors Prof. Jaume Roset and Prof Vojko Kilar for their support and tutoring during the whole process of writing that led to a mark of 8 over 10 at the UPC Physics Department.

The following is an excerpt of the content of the document that will hopefully soon be published in the Physics department's webpage (http://fcia.masters.upc.edu/TFM/llista):

**VARIATIONAL MECHANICS AND NUMERICAL METHODS FOR STRUCTURAL ANALYSIS**

**Abstract**

This work focuses on the particular application of the variational principles of Lagrange and Hamilton for structural analysis. Different numerical methods are compared in their computation of the elastic energy through time.

According to variational mechanics, the difference between the stored elastic energy and the applied work should be null on each time step, so by computing this difference we can account for the level of accuracy of each combination of numerical methods. Moreover, in some situations when numerical instabilities are difficult to perceive due to high complexities, this procedure allows for the control and straightforward visualization of them, being an excellent source of hindsight on the behaviour of the analysed system.

The purpose of this dissertation is to present a scheme where the current numerical methods can be benchmarked in a qualitative as well as in a quantitative manner. It is shown how different combinations of methods, even for a simple model, can give very different results, particularly in the field of dynamics, where often also instabilites arise.

The first half of the thesis is a thorough explanation of these concepts and their application in terms of structural analysis. In the second part, a review on the numerical methods in general and of those implemented for our experiments is provided, followed by the experimental results and their interpretation. The model of choice, for simplicity and availability of analytical results is one cantilever column. Bending elastic energy of the column is monitored under transient regimes of different shapes, computing the total action of the system as its integral through time.

Introduction

Introduction

Targets and interest of our research

Variational mechanics date back as far as the XVIII century, when Leibniz, Euler, Maupertuis and eventually Lagrange devised the calculus of variations and the principle of least action. This methodology of treating physical phenomena is based on the notion that everything in Nature tends to a state of minimal energy(1).

In this work we will be focusing on its particular application in structural analysis, where one deals with “engineering scales” whose dimensions span between 100 times bigger or smaller than those of a human being. This is in contrast with other areas of applied physics like astronomy or molecular dynamics but will be shown how those variational principles still apply and even become powerful tools for the comprehension of the behaviour of our built environment.

Numerical methods, on the other hand, have proliferated since the 1950s alongside with the ever increasing power of computers as a means to simulate physical phenomena. This ceaseless growth in number and terminology has given place to a cumbersome mix of mathematical, physics and computer science often difficult to grasp.

Choosing one simple cantilever beam as our test model, we will utilize and compare different combinations of these methods to compute its elastic energy under transient loading regimes of different shapes. This will render useful in future research in nonlinear analysis of more complex structural systems.

Variational mechanics

According to the principles of variational mechanics (2) , the difference between the measured energy and the applied work should be minimal, so by accounting this difference in each time step of our simulations we should be able to infer the degree of accuracy provided by each combination and discuss the reasons that lead to differences in result using the energy as the natural norm for analysing the error(3).

Numerical methods for structural analysis

In a previous work by the authors (4)(5), it was shown how the vast amount of existing numerical methods can be grouped into three main sets according to the kind of physical phenomena they model and the type of differential equations they discretize: matter integration techniques (Partial Differential Equations), constraint integration techniques (Algebraic Differential Equations) and time integration techniques (Ordinary Differential Equations).

According to this, we will particularize in the following matter integration implementations: Finite Element (FEM), Finite Differences (FDM) and Mass Spring Systems (MSS). For the constraint integration we will be comparing Penalty Method (PM) and Lagrange Multipliers (LM). And for the time integration techniques we will employ Newmark Beta (NB), Houbolt's (HBT), and the Linear Acceleration Method (LAM).

Other combinations are also possible, as the proposed scheme is easily extensible, but for our current purposes it should suffice.

**Discussion and future work**

A numerical comparison of methods commonly employed in structural mechanics was presented.

It was made on the basis of energy principles and eventually the total action of a system under transient loading has been computed for each possible combination of methods.

It was shown how variational principles and an energetic norm can be employed in the benchmarking and assessment of the accuracy and stability of different implementations.

The scheme provided, tested on a simple example, is easily extensible to more complex systems with more elements. The advantage of this approach is that it allows for the monitoring of the global behaviour by means of one simple scalar, whose value is to be compared against that of an analytical computed from external forces or accelerations.

Also, a conceptual framework for the classification and treatment of numerical methods, grouping them into time, matter and constraint integrators, was used for the systematic analysis of the results.

Future work aims at the application of the same methodology in nonlinear analysis and more complex structures.

The combination with stochastic techniques for the integration of the action and the search of minimal energy states is one of the final targets of the current research.

**References**

1) C. Lanczos, 1970, “The variational principles of mechanics”, University of Toronto Press, Canada

2) W. Wunderlich, W Pilkey, 2002. “Mechanics of structures. Variational and computational methods”, CRC Press, pp.: 852-877

3) G. Bugeda, 1991, “Estimacion y correccion del error en el analisis estructural por el MEF”, Monografia 9, CIMNE, Barcelona

4) R. Andujar, J. Roset, V. Kilar, 2011. “Beyond Finite Element Method: An overview on physics simulation tools for structural engineers“, TTEM 3 / 2011. BiH.

5) R. Andujar, J. Roset, V. Kilar, 2011. “Interdisciplinary approach to numerical methods for structural dynamics”, WASJ Vol 14 Num.8, 2011. Iran.

6) J.H. Argyris, 1960, “Energy theorems and structural analysis”, Butterworths Scientific publications, London, UK.

7) R. Aguiar, 2005, “Analisis estatico de estructuras”, Colegio de ingenieros civiles de Pichincha, Quito, Ecuador.

8) H.P. Gavin, 2012, “Strain energy in linear elastic solids”, Duke University, Department of Civil and Enironmental Engineering, USA.

9) A. Nealen, M. Müller, R. Keiser, E. Boxerman, M. Carlson, “Physically Based Deformable Models in Computer Graphics” in Computer Graphics Forum, Vol. 25, issue 4, 2005

10) T.M. Wasfy, A.K. Noor, “Computational strategies for flexible multibody systems” in Appl. Mech. Rev. vol 56, no 6, Nov 2003

11) G.R. Liu, 2003, “Mesh free Methods: Moving Beyond Finite Element Methods”, CRC Press, USA.

12) J. S. Przemieniecki, 1968, “Theory of matrix structural analysis”, McGraw-Hill. Inc, USA

13) G.D. Smit, 1978, “Numerical Solution of Partial Differential Equations by Finite Difference Methods”, 2nd ed. Oxford Applied Mathematics and Computing Science Series, UK.

14) M. Müller, 2008, “Real Time Physics course notes”, Siggraph USA.

15) K.J Bathe, 1995, “Finite element procedures in engineering analysis”, Prentice Hall, USA.

7. Key words

Finite Element, Finite Differences, Variational mechanics, Euler-Bernoulli beam