This introduction concludes with an outline of the individual chapters in order to pro-vide an overview of the subsequent matters and the structural setup of this thesis.
In Chapter 2, the governing equations of differential geometry and non-polar contin-uum mechanics of solids are presented, which yield a set of coupled partial differential equations. The strong form of the field equations of solid continua are subsequently re-formulated by means of variational methods. For the formulation of displacement-based isogeometric shell finite elements, the principle of virtual work is used, whereas the two-field Hellinger-Reissner principle forms the variational basis for a mixed displacement-stress formulation to remove locking.
Chapter 3 introduces the topic of NURBS, which represent the standard function type used in engineering design and which, for analysis, will be applied to both shell geometry representations and discretizations of the solution functions. Therefore, the concepts of knot vectors, patches, control polygons and projective geometry are introduced. Fur-thermore, several strategies for a systematic modification of the NURBS basis and thus the geometric object are presented. These are also utilized to control the discrete solu-tion spaces of physical quantities.
By taking into account the derivations of Chapter 2 for solid continua, a new hierarchic family of 3-, 5- and 7-parameter shell models is developed in Chapter 4; see alsoEchter et al.(2013). Starting from a basic shell model with three independent parameters which mechanically represents the Kirchhoff-Love kinematics, additional parameters are systematically added to improve the approximation quality of the shells. The procedure of adding the extra degrees of freedom is performed so that the kinematic equations of the basic model are gradually enhanced to generate the 5- and 7-parameter models, without the need of a completely new description of the shell kinematics. Additionally, 5- and 7-parameter shell models with non-hierarchic difference vector formulation are established for reasons of comparison.
Chapter 5 covers the discrete formulation of the shell equations of Chapter 4 by using the NURBS function definitions of Chapter 3 in an isogeometric concept. The result-ing algebraic equations represent pure displacement formulations. Besides investigatresult-ing the effect of higher-order and higher-continuity NURBS discretizations on the accuracy of the discrete solution functions, a systematic analysis with regard to locking of the displacement-based NURBS shell elements of this thesis is carried out.
The numerical experiments of Chapter 5 reveal that, along with an improved accuracy of the higher-continuity NURBS shell discretizations, the membrane part of the hierarchic shell elements with pure displacement ansatz is considerably prone to locking. There-fore, in Chapter 6, two methods are developed which successfully remove geometric locking from the in-plane part of the shell finite elements of this work for both higher-order and higher-continuity NURBS discretizations (Echter and Bischoff(2010), Echter et al.(2013)). The first approach generalizes the discrete strain gap (DSG) method ofKoschnick et al.(2002) andBischoff et al.(2003) to higher-continuity NURBS, whereas the second ansatz relies on a mixed displacement-stress formulation based on a two-field Hellinger-Reissner variational principle. The resulting hierarchic NURBS shell elements with modification of the membrane part are completely free from any geometric locking effects.
In Chapter 7, more complex numerical examples with multiple NURBS patches are analyzed in order to investigate the efficiency of the new isogeometric shell element for-mulations.
Chapter 8 provides a summary of the developments in the thesis along with conclusions and indications of future work.
In Appendix A.1, further mathematical background on vector and tensor algebra fun-damentals is presented.
2
Continuum Mechanics, Differential Geometry
In this chapter, the governing equations of differential geometry and non-polar contin-uum mechanics of three-dimensional solids are presented to an extent required within this thesis and to establish a uniform notation. The balance laws used in this work are confined to mechanical problems and serve as a basis for the material-independent formulation of the deformation processes of solids.
First, the fundamentals of elementary differential geometry required to describe ge-ometric configurations of deformable solid bodies in space are introduced. Without considering microscopic physical effects, the principal equations of motion and deforma-tion, i.e. the kinematics of continuous solid media are defined in material (Lagrangian) description.
Based on the concepts of configuration and motion of continuous bodies, appropriate strain measures and strain-displacement relations are established as one of the field equations of continuum mechanics.
The interaction of material within a body during deformation results in the notion of a stress state. By applying Cauchy’s stress theorem, surface tractions are uniquely related to second-order stress tensor fields. Additionally, relevant alternative stress tensors are defined.
From the momentum balance laws and Cauchy’s stress theorem, the equilibrium equa-tions of elastostatics as the second field equation are subsequently derived.
The third functional field equation of continuum mechanics to be introduced is the con-stitutive law in order to sufficiently describe the response of a material and to specify the stress-strain relationship.
Finally, the governing equations which consist of kinematics, equilibrium and material
law are reformulated by means of variational and energy principles. They serve as a basis for the development of numerical approximation and discretization methods as the finite element method to be used within this thesis.
A more comprehensive treatment of the basics of both differential geometry and con-tinuum mechanics for solids and structures is covered, for instance, inHsiung(1981), Kreyszig(1991),Ciarlet(2006),Marsden and Hughes(1983),Ciarlet(1988), Altenbach and Altenbach (1994), Stein and Barthold (1996), Holzapfel (2000),Zienkiewicz et al.(2005).
2.1 Elementary differential geometry
Differential geometry which relates the mathematical branches of analysis and geome-try enables an elegant analytic investigation of the geometric properties of point sets by using methods of infinitesimal, i.e. differential and integral calculus.
For all quantities and relations introduced to describe the configuration and motion of a material bodyBin 3D Euclidean spaceR3, the concept of classical differential geometry is applied. A material bodyBin continuum mechanics corresponds to a contiguous set of material pointsM. Its boundary is denoted with∂B.
An orientation in spaceR3is defined with the introduction of a fixed orthogonal Carte-sian coordinate system. Its orthonormal basis ei is pointing in the direction of the coordinate axes and the reference point0is coincident with the origin of the Cartesian basis. Thus the position of every spatial pointP relative to the origin 0is uniquely associated with a position vectorXas the linear combination of material convective coordinatesXiand base vectorsei, as shown in Figure 2.1. In this work, Latin indices run from 1 to 3 and Greek indices take on the values 1 or 2.
X=Xiei, with ei=ei,∥ei∥= 1 (2.1)
The symbol∥(•)∥in Equation (2.1) denotes the Euclidean norm of (•) inR3. Conse-quently a bijective, i.e. continuous and invertible mapτ, which in mathematical litera-ture is frequently denoted as homeomorphism, may be defined. It uniquely associates to every material pointM∈Bfor every point in time a corresponding pointPin Euclidean spaceR3labeled with a position vectorX. For a fixed moment in time, the mapping τofBto its imageΩinR3represents a configuration which is specified in Equation (2.2)
τ:
⎧
⎨
⎩
B→Ω⊂R3
M+→X=τ(M). (2.2)
The inverse mapping of spatial points ofR3to the material pointsM∈Bexists as well and is denoted withτ−1. Moreover, convective curvilinear coordinates are introduced in order to adequately describe the geometry and kinematics of the curved structural objects in this thesis. Therefore each pointP∈R3can be uniquely identified by the co-ordinatesθi. The position vectorX, which in Equation (2.1) is related to the Cartesian basiseiin curvilinear convective coordinates reads
X=X&θi'. (2.3)
The partial derivative ofXwith respect toθiyields the covariant base vectorsGi, which represent tangents to the coordinate lines θi and in contrast toei, are in general not orthonormal. The expression (•),k corresponds to the partial derivative of a quantity (•) with respect to the argumentk.
Gi=X,i=∂X
∂θi =∂Xj
∂θi ej (2.4)
The base vectors Gi define a tangent vector space at every pointP and enable the derivation of important local geometric properties to be specified later in this section.
M
B P
Ω
e1
e2 e3
G1 G2
G3
θ1
θ2 θ3
X
0 τ τ−1 Material body
Configuration
Figure 2.1:Mappingτof material pointsMto spatial pointsP.
The contravariant base vectorsGimay be consequently derived by Gi= ∂θi
∂Xjej, Gi·Gj=δij. (2.5)
They represent a dual basis toGi. The Kronecker deltaδij takes the value of one if i=jand is zero otherwise.
With the co- and contravariant base vectors at hand, the components of the metric tensorGcan be directly computed according to Equation (2.6)
G=Gi·GjGi⊗Gj=GijGi⊗Gj=GijGi⊗Gj. (2.6) Thus, important local geometric quantities with regard to the metric of the body, such as distances between points, lengths or angles between tangent vectors, or the size of differential line, area or volume segments may be derived.
The symbol (⊗) in Equation (2.6) represents the tensor product of the respective base vectors in either covariant or contravariant form. The contravariant components of the metric tensor are obtained by the inverse ofGij. They enable the calculation of the contravariant base vectorsGi, which also lie in the tangent space that is spanned by the covariant basis.
Gi=GijGj (2.7)
The kinematic equations of the isogeometric shell models of Chapter 4 only require the definition of the metric tensorGwithout explicitly using curvature tensor properties as-sociated with the second fundamental form of differential geometry, which will therefore be omitted in the subsequent derivations. For more information on curvature character-istics of curves and surfaces referencesKlingbeil(1966),Başar and Krätzig(1985) andCiarlet(2006) are recommended.