# Tensor-based matrices in geometrically non-linear FEM

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2005; 63:20862101Published online 3 May 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1343

Tensor-based matrices in geometrically non-linear FEM

V. V. Chekhov,

Yalta Management University, 8, vul. Rudanskogo, Yalta, 98600 Crimea, Ukraine

SUMMARY

In the framework of the object-oriented paradigm, advantages of using the index-free tensor notationin combination with the concept of generalized tensor-based matrix are considered as being the mostcorresponding to the paradigm. The advantages reveal itself in the disappearance of a semantic gapbetween various stages of creation of FEM applications (theoretical inferences, use of the numericalmethods, object-oriented software implementation) and, as a result, use of the unied object modelin all the stages, as well as simplication of theoretical transformations. Based on the consideredapproach, a new FEM equation for large strain analysis is developed and its solution technique isoutlined. Copyright 2005 John Wiley & Sons, Ltd.

KEY WORDS: FEM equation; large strain; object-oriented approach; tensor-based approach

INTRODUCTION

The object-oriented (OO) approach [1], intended for investigation, modelling and developmentof complex systems, has found numerous realizations in the theory and practice of the niteelement method (FEM) at present time. In particular, the use of OO approach proves to be ratherefcient in the eld of structural analysis for the geometrically non-linear computations [29],when complexity of the FEM concept is intensied by the complexity of non-linear mechanicalformulations. A semantic gap between tensor form (natural for non-linear solid mechanics) andthe matrix one (conventional for FEM and convenient for manipulation by standard techniquesof matrix algebra) represents rather vexed problem (in comparison with linear analysis). Inthe majority of publications (where the subject is obviously mentioned) this gap is overcomeby introducing a transformation between the tensor and matrix forms. Such transformationcan be made in deducing of equations (then only matrix objects are available in the objectmodel and tensors are presented implicitly by their components, e.g. in Reference [4]), or inthe program implementation when the object model contains both matrix and tensor layouts,e.g. in Reference [8]. Both methods need major intellectual and/or computational effort since,

Correspondence to: V. V. Chekhov, 14-11, vul. Shpolyanskoy, Simferopol 95034, Ukraine.E-mail: v_chekhov@ukr.net

Received 23 May 2004Revised 26 September 2004

Copyright 2005 John Wiley & Sons, Ltd. Accepted 31 December 2004

TENSOR-BASED MATRICES IN GEOMETRICALLY NON-LINEAR FEM 2087

actually, they do not remove the gap. In Reference [5] tensor and matrix classes are declared(by means of inheritance) as kinds of the concept of a multidimensional array; as a result, thegap is removed. The presented paper is also devoted to elimination of the above-mentioned gapby introducing for tensors a uniform representation to be used both for theoretical developmentand for practical implementation. An approach similar to the one used in References [10, 11] isapplied. In particular, its advantages, expressing in simplication of deducing new theoreticalresults for non-linear FEM, are demonstrated in this work.

INVARIANT TENSOR OBJECTS

Though, practically each research devoted to the OO FEM implementation introduces its ownobject model distinguished from other models developed by other authors (since the choiceof what components in a system are primitive is relatively arbitrary and is largely up to thediscretion of the observer of the system [1]), nevertheless it is possible to pick out a numberof the standard objects that belong to the problem domain (nite elements, nodes, matrices,etc.), which are present practically in all its OO implementations. In particular, ones that belongto the most important objects for geometrically non-linear consideration are tensors.

Besides matrix form which is especially popular and present practically everywhere in FEMapplications, there are two more forms for representation of tensors in the state-of-the-arttensor calculus [12]. Component form, which looks like a value having indices, is used, e.g.in References [46]; moreover, in the latter two this form is the base of the overall objectmodel. In the index-free form, which is developed later than others, tensor has no free indicesand, nominally, it is a contraction of its components with corresponding vectors of a basis (e.g.A=Asteset ). This fact allows us to write down mechanical relations in highly compact andindependent from basis form [13, 14], then indices do not occult physical essence of laws.Such form is used in References [79] (alongside with matrix one) as a part of the objectmodel for non-linear FEM.

For comparison of the above-mentioned forms from the OO point of view, it is necessaryto note that, conceptually, tensor is an invariant object in the space, which does not dependon selection of a co-ordinate frame and basis. Though, this is valid in all formulations, forthe component form, only a part of a tensor (components in some concrete basis), which isnot invariant, is formally used as the tensor. At the same time we have to keep inwardly theinvariance of the tensor, and the responsibility for its correct treatment is up to the user. The OOapproach must unload the person and shift the responsibility for implementation of an object tothe object. From this viewpoint, the matrix form is more convenient. However, in the theory, itis used as auxiliary one and does not quite meet the tensor paradigm: tensor is not a matrix yetsince it contains innitely many matrices (co- and contravariant components for all possibleco-ordinate frames). A matrix or a multidimensional array may be internal implementation ofa tensor (the tensor may be asked for it) but it must not represent the interface of the tensor.The mentioned drawbacks are absent in the index-free form where all indices and basis vectorsare hidden within the object and, actually, are details of its internal implementation. This justcorresponds to the OO paradigm. Due to extension of mathematical functions, the differentialand integral operator [12] handling of tensor objects becomes as evident as that with scalarreal quantities. Such objects are used as the unied representation of tensors and as the basefor the object model in References [10, 11]. Although these papers are not devoted solely to

Copyright 2005 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2005; 63:20862101

2088 V. V. CHEKHOV

the non-linear FEM, they justly assert that benets of the used approach can also be utilizedfor non-linear problems. In this manner, the presented work can be considered as an extensionof this approach to the area of geometrical non-linearity.

Non-linear theory of elasticity in index-free form is very compactly stated in Reference [14].The present paper in notations bases substantially in this source together with References [13, 15].For example, such tensors as position vector, Hamiltons nabla operator (rst-rank tensors),identity (metric) tensor (second-rank tensor) look like

r= r(qs), = es qs

, 1= st eset = eses = eses

where qs are co-ordinates, es = rqs are vectors of a current basis, es are vectors of the reciprocalone. First and second order tensors are denoted by bold face, higher order tensorsby boldface with explicitly indicated order at left superscript. Scalar product, double scalar product,etc. are denoted by appropriate number of dots. Direct product is denoted by no characters.For example, double dot product for second-order tensors has the next behaviour

p q=q p=pT qT =qT pT (1)

Extraction of a scalar from a tensor is a standard technique to achieve commutativity in longformulae

a= 1 a= eses a= es(es a) (2)

Based on the denition of derivation of tensors with respect to a tensor argument [12, 14],rules for derivation of composite functions with respect to a rst-order tensor can be written as

(p)a = pa + pa(p q)a = (pa es) qes + p qa(p q)a = (pa es) qes + p qa(p1)a = p1 (pa es) p1es

(3)

Index-free form of mechanical formulationsWhile considering geometrical non-linearity, it is necessary to distinguish an initial congurationof a body and the current conguration. Therefore all quantities relating to the former aredenoted with zero above. Tensors of the second and higher orders whose different indicescorrespond to different congurations (so-called double tensors [15]), are denoted by sequencesof 0 and t . These sequences indicate which conguration respective index refers to (0initial,tdeformed). Quantities relating to the local frame of a nite element are denoted with above. Below the required values and formulations of the theory of elasticity are given in theindex-free form.

Copyright 2005 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2005; 63:20862101

TENSOR-BASED MATRICES IN GEOMETRICALLY NON-LINEAR FEM 2089

Deformed state of a continuum is determined by the displacement vector u= r 0r and thetensor

0r= 0eses = 1 +0u. Polar decomposition of the deformation gradient

0r= 0 0tO= 0tO (4)

denes an orthogonal tensor0tO specifying rotation of a material particle, and tensors

0=

0r ( 0r)T and = 0tOT 0 0tOleft (Lagranges) and right (Eulers) stretch tensors, respec-

tively.Strain measures: the linear strain tensor

0= 12 (

0u + ( 0u)T)= 12 (0r + ( 0r)T) 1 (5)

the Green (-Lagrange) strain tensor0E= 0 + 12

0u ( 0u)T = 12 (0r ( 0r)T 1)= 12 (

02 1) (6)

the Green strain tensor increment after cutting term

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