A Unified Finite Strain Theory for Membranes and Ropes
AA Unified Finite Strain Theory for
Membranes and Ropes
T.P. Fries, D. SchöllhammerSeptember 30, 2019
Institute of Structural AnalysisGraz University of TechnologyLessingstr. 25/II, 8010 Graz, Austria
Abstract
The finite strain theory is reformulated in the frame of the Tangential DifferentialCalculus (TDC) resulting in a unification in a threefold sense. Firstly, ropes, mem-branes and three-dimensional continua are treated with one set of governing equa-tions. Secondly, the reformulated boundary value problem applies to parametrized and implicit geometries. Therefore, the formulation is more general than classicalones as it does not rely on parametrizations implying curvilinear coordinate systemsand the concept of co- and contravariant base vectors. This leads to the third uni-fication: TDC-based models are applicable to two fundamentally different numericalapproaches. On the one hand, one may use the classical Surface FEM where the geom-etry is discretized by curved one-dimensional elements for ropes and two-dimensionalsurface elements for membranes. On the other hand, it also applies to recent TraceFEM approaches where the geometry is immersed in a higher-dimensional backgroundmesh. Then, the shape functions of the background mesh are evaluated on the trace ofthe immersed geometry and used for the approximation. As such, the Trace FEM is a a r X i v : . [ c s . C E ] S e p fictitious domain method for partial differential equations on manifolds. The numeri-cal results show that the proposed finite strain theory yields higher-order convergencerates independent of the numerical methodology, the dimension of the manifold, andthe geometric representation type.Keywords: finite strain theory, ropes, membranes, Trace FEM, fictitious domainmethod, embedded domain method, PDEs on manifolds ONTENTS Contents . . . . . . . . . . . . . . . . . . . . . 142.3.2 Manifolds with codimension . . . . . . . . . . . . . . . . . . . . . 172.4 Similarities and differences in the parametric and implicit descriptions . . . 182.5 Further definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.1 Covariant surface gradient and divergence . . . . . . . . . . . . . . 212.5.2 Conormal vector on the boundary . . . . . . . . . . . . . . . . . . . 212.5.3 Divergence theorem on manifolds . . . . . . . . . . . . . . . . . . . 22 CONTENTS
In structural analysis, there are many examples where a dimensional reduction in the mod-eling is a key step to achieve simpler models than considering the full (three-dimensional)continuum. The situation is simple for flat domains such as straight beams and flat plates.For curved structures with reduced dimensionality such as curved beams and shells, model-ing becomes considerably more involved and the governing equations are partial differentialequations (PDEs) on manifolds. The situation is even more complicated for structures un-dergoing large displacements such as ropes (cables) and membranes. These structures donot feature any bending resistance and often deform largely such that one has to carefullydistinguish between the undeformed and deformed configurations. When equilibrium isto be fulfilled in the deformed configuration, this is generally referred to as finite straintheory, the geometrically non-linear situation, or large displacement analysis.Herein, we propose a new formulation of the mechanical models for ropes and membranesin finite strain theory which employs the Tangential Differential Calculus (TDC) for thedefinition of geometric and differential quantities. The geometries of ropes and membranesmay be seen as (curved) manifolds embedded in a higher-dimensional space. Let thedimension of the surrounding region be d and the dimension of the manifold be q with q = 1 for ropes and q = 2 for membranes. The classical modeling approach to ropes andmembranes (including shells) may be called parameter-based [9], because it relies on aparametrization which maps from the q -dimensional to the d -dimensional space. In thiscase, a curvilinear coordinate system is naturally implied and co- and contra-variant basevectors are easily defined. An alternative is to define the geometry implicitly, for example,based on the level-set method [36, 35, 46]. Then, the zero-level set of a scalar functionimplies the manifold. In this case, no parametrization, hence, no curvilinear coordinatesexist on the manifold. This situation cannot be covered by the classical models but theproposed TDC-based approach can.When comparing the new TDC-based formulation with the classical parameter-based for-mulation, we notice the following important aspects:1. The first argument is geometrical and is concerned with the definition of the boundaryvalue problem (BVP) of ropes and membranes. It is desirable to have a mechanicalmodel which applies to general geometries no matter whether they are defined inparametric or implicit form. The classical parameter-based approach does not apply Introduction to the case where the geometry is, for example, defined by a zero-level set. Inthat case, a parametrization only becomes available upon meshing the zero-levelset (resulting in an atlas of mappings implied by the elements). However, thenthe BVP is rather defined with respect to a discrete setting than a continuous one.Consequently, one advantage of the TDC-based formulation is that it allows for aproper definition of a continuous BVP which applies for both, parametrically andimplicitly defined geometries.2. The second argument is mechanical . The TDC-based formulation treats ropes andmembranes in a unified sense where all mechanical quantities such as stress andstrain tensors are based on differential operators formulated with the TDC. Thesequantities are defined based on the d -dimensional global coordinate system into whichthe manifold is immersed. That is, these tensors are always d × d no matter whetherropes or membranes are considered. The TDC-based formulation may also be seen asa special case of a d -dimensional continuum (non-manifold case) where the manifold-operators become classical ones. In contrast, in the parameter-based formulation,where the q -dimensional manifold results from mapping q parameters to d dimensions,the related mechanical tensors are q × q and thus inherently different for ropes,membranes and continua [9]. For the TDC-based situation, these tensors result fromthe same formulas, however, based on different differential operators depending onthe mechanical situation.3. The third argument is numerical and related to discretization methods. The newformulation allows for two fundamentally different numerical approaches. One maybe seen as the classical approach which relies on meshing the ropes and membranes by(curved) line or surface meshes; this is called the Surface FEM herein. It is inherentlylinked to the parameter-based formulation of the mechanical models. Due to the factthat for implicit geometries, one may also provide surface meshes, and then continuewith the parameter-based formulation, there was not really a strong need for a moregeneral formulation of the models until recently. However, there is an alternativeapproach to solve ropes and membranes which uses d -dimensional, non-matchingbackground meshes to approximate the displacements instead of conforming, curved q -dimensional surface meshes. This approach has been labeled Trace FEM becausefor the integration of the weak form, the d -dimensional shape functions are onlyevaluated on the trace of the manifold [32, 39, 23]. The Trace FEM is inherentlylinked to implicitly defined manifolds and does not need any parametrization. Thesolution of BVPs based on non-matching background methods is generally referred toas fictitious (or embedded) domain methods (FDMs) with a large number of variantsexisting today. Recently, the Cut FEM has emerged as a popular FDM allowing forhigher-order accuracy [4, 5, 6]. When using the Cut FEM for the solution of PDEson manifolds as done herein, the method becomes analogous to the Trace FEM.Consequently, the TDC-based formulation allows for a unification in a geometrical (para-metric and implicit geometries), mechanical (ropes, membranes, and d -dimensional con-tinua) and numerical (Surface and Trace FEM) sense. For similar reasons, the authorshave already used the TDC-based approach to reformulate the mechanical models of linearKirchhoff-Love [44, 43, 42] and Reissner-Mindlin shells [45]. Using the TDC for the defi-nition of BVPs on manifolds as discussed herein is already well-accepted in the definitionof transport phenomena on manifolds where it has replaced the classical parameter-baseddefinition in many cases [15, 17, 29]. It is thus coming timely and naturally to reformulate mechanical BVPs on manifolds based on the TDC in the same sense.The classical theory of large displacement membranes based on curvilinear coordinates isdescribed, e.g., in [2, 8, 10, 9]. The general equivalence of models using parametrized orimplicit manifolds is outlined in [13]. Geometrically exact shell models based on explicitlydefined surfaces and locally using Cartesian coordinates are defined in the linear setting in[47] and for the non-linear case in [28]. Local Cartesian coordinates are also used in [37],where the initial configuration is modeled with a stress-free deformation of a flat surface.Another approach for explicitly defined geometries is the degenerated solid approach [1].Concerning a TDC-based modeling of large deformation membranes with the Surface FEM,we emphasize the work in [27]. A TDC-related approach in the field of composite structuresis presented in [30] for embedded membranes using complex material models and with focuson analytical solutions. In contrast, herein, the novelty is in the unified treatment of ropes and membranes and the numerical treatment with Surface and
Trace FEM. In this work,we also discuss in detail how ropes and membranes (in two and three dimensions) aregenerally defined using the parametric and the implicit approach, including all relevantgeometric and differential quantities. For example, implicitly defining a rope in threedimensions requires two level-set functions whereas one level-set function is sufficient formembranes. Furthermore, the mechanical discussion presented herein includes stress andstrain tensors, so that the manifold versions of the first and second Piola-Kirchhoff stresstensors, the Cauchy stress tensor, the Euler-Almansi and Green-Lagrange strain tensors
Introduction are explicitly given.Concerning the numerical approximation of ropes and membranes based on the Surface andTrace FEM as discussed herein, it is mentioned that the Trace FEM just as any other FDMsimplifies the meshing of geometries significantly. However, additional effort is required for(i) the integration of the weak form, wherefore suitable integration points on the manifoldshave to be provided [18, 34, 19, 21], (ii) the application of boundary conditions which ismore involved because the boundary is within the background elements and instead ofprescribing nodal values one may have to use Nitsche’s method [6, 3, 26, 41, 40] or otherapproaches for enforcing constraints, and (iii) stabilization terms which are necessary toaddress the existence of shape functions with small supports on the manifolds and tofind unique solutions with the background mesh although the BVP is only defined on themanifold [32, 24, 7, 25]. In spite of the increased effort for implementing the Trace FEM,it may have significant advantages over the Surface FEM. For example, when ropes and/ormembranes are reinforcing sub-structures embedded in some three-dimensional continuum,there is no need to consider these sub-structures in the meshing of the volume. This isthe first work where Trace FEM results for ropes and membranes are shown, enabling theimplicit analysis of these structures.The paper is organized as follows: In Section 2, the parametric and implicit geometry def-initions of manifolds are discussed in detail for the situation of ropes and membranes. Asusual in finite strain theory, the undeformed and deformed situations are distinguished, re-lated by the sought displacements. In each configuration, normal vectors, surface stretches,and differential surface operators are defined, wherefore it is important to distinguish para-metric from implicit definitions. The mechanical modeling is outlined in Section 3 followingthe classical steps defining stress and strain tensors and imposing equilibrium in the de-formed configuration. The strong and weak forms of the governing equations are given.It is noteworthy that ropes and membranes are treated in the same manner and only thegeometry-dependent surface operators defined in Section 2 differ. The numerical solutionof the governing equations is considered in Section 4 where the discrete weak forms of theSurface and Trace FEM are given. The numerical results in Section 5 confirm that bothnumerical approaches achieve higher-order convergence rates. It is also confirmed howropes and membranes may easily be coupled with the presented formulation. The paperends in Section 6 with a summary and conclusions.
The focus of this work is on ropes (also called cables) and membranes undergoing largedisplacements which is covered by finite strain theory. Only solids according to the SaintVenant–Kirchhoff material model are considered herein which may be seen as the sim-plest extension of a linear elastic material model to large displacements. Cables may bemodeled as one-dimensional lines in the two- or three-dimensional space. Membranes aretwo-dimensional surfaces in the three-dimensional space. Hence, we may state that cables and membranes are curved manifolds with a lower dimension q than the surrounding spacewith dimension d . In this section, we address the issue of how to define such manifolds geo-metrically and formulate (differential) operators needed later in the mechanical model, seeSection 3. For the geometry definition, we separately discuss the situation for parametrizedand implicit manifolds. Implicit manifolds are implied by level-set(s) of scalar function(s)following the concept of the level-set method [36, 35, 46]. For the parametric situation, theoutline is related to the classical setup using tensor notation rather than index notationand avoiding any explicit reference to classical terms in the context of curvilinear coor-dinate systems such as co- and contra-variant base vectors and Christoffel symbols. Forthe implicit situation, the presented outline, systematically including all geometric anddifferential quantitites for ropes and membranes in finite strain theory, is original. As usual in finite strain theory, we consider an undeformed material configuration and adeformed spatial configuration. These are represented by the q -dimensional manifolds Γ X and Γ x , respectively, which are immersed in a d -dimensional space R d , herein, d = { , } .The difference d − q is also called codimension of the manifold. We follow the usualnotation to relate uppercase letters in variable and operator names with the undeformedconfiguration and lowercase letters with the deformed one. The displacement field u ( X ) relates the two configurations via x = X + u ( X ) with X ∈ Γ X ⊂ R d and x ∈ Γ x ⊂ R d . Tangential differential calculus in finite strain theory (a) reference domain Ω r (b) def. and undef. configuration, Γ X and Γ x Figure 1: The situation for membranes given by a parametrization: (a) The referencedomain Ω r ⊂ R , (b) the undeformed domain Γ X resulting from a given parametrization X ( r ) and the deformed domain Γ x resulting from the deformation x = X + u . For parametrized manifolds, there exists a map X ( r ) : R q → R d from some lower-dimensional reference domain Ω r ⊂ R q to the undeformed configuration Γ X ⊂ R d . Animportant consequence is that local curvilinear coordinate systems result naturally on themanifolds. It is useful to describe the situation separately for cables (one-dimensionalmanifolds) and membranes (two-dimensional manifolds). We start with two-dimensional manifolds ( q = 2 ) in the three-dimensional space ( d = 3 )which is relevant for membranes. Let there be a reference domain Ω r ⊂ R and a map X ( r ) : Ω r → Γ X ⊂ R , see Fig. 1. We label the components r = [ r, s ] T and X =[ X, Y, Z ] T . The Jacobi matrix J ( r ) = ∂ X ( r ) ∂ r = ∇ r X ( r ) = ∂ r X ∂ s X∂ r Y ∂ s Y∂ r Z ∂ s Z .2 Parametrized manifolds (3 × . One may easily obtain two vectors T (cid:63) = ∂ X ∂r and T (cid:63) = ∂ X ∂s fromthe columns of J being tangential to Γ X at a mapped point X . The Jacobi matrix isalso used to compute the first fundamental form G and the operator Q , later needed fordefining surface gradients, G = J T · J ( q × q ) -matrix , Q = J · G − ( d × q ) -matrix . (2.1)Next, we consider a displacement field u ( r ) assuming that a point r is given which mayalso be seen as a function u ( r ( X )) when a point X ∈ Γ X is given (and back-projectedto the reference domain inverting the map X ( r ) ). We emphasize that in both cases, thedisplacement field only lives on the manifold Γ X . Hence, no classical partial derivativeswith respect to X may be computed (unless u is smoothly extended to some neighborhoodof X which is not unique and not considered here) so that the only useful gradient of u inthis context is the surface gradient . For some scalar function f ( r ) , e.g., each displacementcomponent, the surface gradient is ∇ Γ X f ( r ) = Q · ∇ r f ( r ) ⇔ ∂ Γ X f∂ Γ Y f∂ Γ Z f = Q · (cid:34) ∂ r f∂ s f (cid:35) , For a vector function u ( r ) = [ u, v, w ] T ∈ R , we have the directional surface gradient ∇ Γ , dir X u ( r ) = (cid:0) ∇ Γ X u (cid:1) T (cid:0) ∇ Γ X v (cid:1) T (cid:0) ∇ Γ X w (cid:1) T = ∂ Γ X u ∂ Γ Y u ∂ Γ Z u∂ Γ X v ∂ Γ Y v ∂ Γ Z v∂ Γ X w ∂ Γ Y w ∂ Γ Z w = ∇ r u ( r ) · Q T , (2.2)which is to be distinguished from the covariant surface gradient of a vector field definedin Section 2.5.1.Let us next consider the map from the undeformed to the deformed configuration x ( X ) = X + u ( X ) which is Γ X → R d . The related Jacobi matrix is also called surface deformationgradient , F Γ = ∇ Γ , dir X x ( X ) = I + ∇ Γ , dir X u ( X ) , (2.3)where I is a ( d × d ) identity matrix.One may obtain all equivalent quantities in the deformed configuration: The Jacobi-matrixfrom the reference to the deformed configuration j = F Γ · J and the tangent vectors t (cid:63) ,2 Tangential differential calculus in finite strain theory t (cid:63) to the deformed configuration Γ x at a mapped point x based on the Jacobi matrix j .Furthermore, the first fundamental form g = j T · j and the operator q = j · g − relating theclassical gradient of the reference configuration with the surface gradient of the deformedconfiguration as ∇ Γ x f = q · ∇ r f .Based on the pairs of tangent vectors in the undeformed and deformed configuration, onemay compute unique normal vectors in each configuration, N (cid:63) = T (cid:63) × T (cid:63) and n (cid:63) = t (cid:63) × t (cid:63) .Then, the projectors P ( X ) and p ( x ) are computed as P = I − N ⊗ N with N = N (cid:63) (cid:107) N (cid:63) (cid:107) , (2.4) p = I − n ⊗ n with n = n (cid:63) (cid:107) n (cid:63) (cid:107) . (2.5)The same result is obtained when computing a tangent vector T (cid:63) in the tangent planespanned by T (cid:63) and T (cid:63) being orthogonal to T (cid:63) using Gram Schmidt orthogonalization, T (cid:63) = T (cid:63) − T (cid:63) · T (cid:63) T (cid:63) · T (cid:63) · T (cid:63) , then P = T ⊗ T + T ⊗ T with T = T (cid:63) (cid:107) T (cid:63) (cid:107) , T = T (cid:63) (cid:107) T (cid:63) (cid:107) , analogously for p . The projector P at some point X maps an arbitrary vector in R d tothe tangent space at Γ X , hence, P · N = . P is symmetric, P = P T , and idempotent, P · P = P , which holds analogously for p .Next, we are interested in the stretch of a differential element of the membrane whenundergoing the deformation. This is interpreted as an area stretch and defined as Λ = √ det g √ det G = (cid:107) n (cid:63) (cid:107)(cid:107) N (cid:63) (cid:107) = (cid:107) t (cid:63) × t (cid:63) (cid:107)(cid:107) T (cid:63) × T (cid:63) (cid:107) . Finally, an operator W is introduced which relates surface gradients of the undeformedand the deformed configuration as ∇ Γ x f = W · ∇ Γ X f with W = q (cid:0) Q T · Q (cid:1) − Q T . (2.6)This result is obtained using ∇ Γ X f = Q · ∇ r f and ∇ Γ x f = q · ∇ r f . Note that Q and q are ( d × q ) -matrices, hence, not quadratic and the concept of generalized inverses (or pseudoinverses) is needed to obtain Eq. (2.6). .3 Implicit manifolds (a) reference domain Ω r (b) def. and undef. config., Γ X and Γ x Figure 2: The situation for cables given by a parametrization: (a) The reference domain Ω r ⊂ R , (b) the undeformed domain Γ X resulting from a given parametrization X ( r ) andthe deformed domain Γ x resulting from the deformation x = X + u . Consider some one-dimensional reference domain Ω r ⊂ R and a map X ( r ) : Ω r → Γ X ⊂ R d , d = { , } , see Fig. 2. This situation applies to cables in two and three dimensions.The Jacobi matrix J ( r ) = ∇ r X ( r ) consists of one column only which implies a tangentvector T ∈ R d being tangential to the undeformed configuration at X . For the tangentvector in the deformed configuration follows t (cid:63) = F Γ · T (cid:63) . Most parts of the discussionfrom Section 2.2.1 apply accordingly. However, the definition of the projectors changes to P = T ⊗ T with T = T (cid:63) (cid:107) T (cid:63) (cid:107) , (2.7) p = t ⊗ t with t = t (cid:63) (cid:107) t (cid:63) (cid:107) . (2.8)For the stretch of a differential element of the cable, which may be seen as a line stretch,there follows Λ = √ det g √ det G = (cid:107) t (cid:63) (cid:107)(cid:107) T (cid:63) (cid:107) . Implicit manifolds are implied by one or more level-set functions. Generally speaking,the codimension determines the number of level-set functions required to define a unique4
Tangential differential calculus in finite strain theory (a) (b)
Figure 3: Two different implicit definitions of the same manifold (blue surface) in R withcodimension : (a) The manifold is defined by one (master) level-set φ ( X ) (red surface)which is bounded by additional (slave) level-set functions ψ i ( X ) (yellow surfaces). (b) Themanifold is defined by one level-set φ ( X ) which is evaluated in the domain of definition Ω X (yellow body).geometry of a manifold. For the cases relevant in this work, this means that one level-setfunction φ ( X ) is required for cables in R and membranes in R which have codimension . For cables in R , on the other hand, two level-set functions φ ( X ) and φ ( X ) areneeded. We split the discussion depending on the codimension. Oriented manifolds with codimension may be defined by one level-set function φ ( X ) .Usually, the zero -level set of φ implies the manifold of interest and there are infinitely manypossible φ implying the same geometry. The signed distance function is a particularly usefulconcrete example for φ ( X ) and often used in practice. It is noteworthy that many level-setsare unbounded in R d which is not desirable for the definition of mechanical applications.Fortunately, it is easily possible to define bounded manifolds by additional (slave) level-setfunctions ψ i ( X ) , see [20, 22]. Then, the undeformed configuration is, e.g., given by Γ X = (cid:8) X ∈ R d : φ ( X ) = 0 , ψ i ( X ) ≥ , i = 1 , . . . n (cid:9) , (2.9) .3 Implicit manifolds R with codimension , e.g., a cable in twodimensions: The undeformed situation is implied by the zero-level set of φ ( X ) (blue line),evaluated in the domain of definition Ω X (colored region). The deformed configurationresults from the displacement field u ( X ) and x = X + u .as shown in Fig. 3(a). Even simpler is to associate a domain of definition Ω X ⊂ R d with φ , then the bounded manifold results as Γ X = { X ∈ Ω X : φ ( X ) = 0 } . (2.10)In this case, the boundaries are the intersections of the zero-level set with the boundaryof the domain of definition, see Fig. 3(b). The implicit definition according to Eq. (2.10)is used in the following unless noted otherwise.Next, we focus on the situation in large displacement theory as shown in Fig. 4 for thisimplicit setup. The normal vector of the undeformed configuration is obtained by thegradient of the level-set function, N (cid:63) ( X ) = ∇ X φ ( X ) for X ∈ Γ X . Let there be a displacement field u ( X ) which lives in the full d -dimensional space (in-stead of only the manifold itself as for parametric manifolds) so that the classical gradient ∇ X u ( X ) is available. The resulting deformation gradient is F Ω = ∇ X x ( X ) = I + ∇ X u ( X ) (2.11)6 Tangential differential calculus in finite strain theory which is different from the surface deformation gradient F Γ in Eq. (2.3). Based on this,one may compute the normal vector of the deformed configuration at x = X + u ( X ) as n (cid:63) ( x ) = ∇ x φ ( X ( x )) = F − TΩ · N (cid:63) for x ∈ Γ x , which follows by the chain rule. Eqs. (2.4) and (2.5) are used to compute the projectors P ( X ) and p ( x ) , respectively. The surface gradient (with respect to the undeformedconfiguration) of a scalar function f ( X ) with X ∈ R d results as ∇ Γ X f = P · ∇ X f. (2.12)As before, ∇ X f is the classical gradient in the d -dimensional space. The situation isanalogous for each component u i of a vector function u ( X ) , so that one obtains for the directional surface gradient ∇ Γ , dir X u = ∇ X u · P , (2.13)for u ∈ R : ∂ Γ X u ∂ Γ Y u ∂ Γ Z u∂ Γ X v ∂ Γ Y v ∂ Γ Z v∂ Γ X w ∂ Γ Y w ∂ Γ Z w = ∂ X u ∂ Y u ∂ Z u∂ X v ∂ Y v ∂ Z v∂ X w ∂ Y w ∂ Z w · P P P P P P P P P . The surface deformation gradient F Γ follows using Eq. (2.3). The stretch of a differentialelement upon the deformation is Λ = (cid:107)∇ x φ (cid:107)(cid:107)∇ X φ (cid:107) · det F Ω = (cid:107) n (cid:63) (cid:107)(cid:107) N (cid:63) (cid:107) · det F Ω . Finally, for the operator W relating surface gradients of the undeformed and deformedconfiguration, one obtains ∇ Γ x f = W · ∇ Γ X f with W = p · F − TΩ . (2.14)Note that W · P = W , hence, when using the classical derivatives one obtains ∇ Γ x f = W · ∇ X f . .3 Implicit manifolds R with codimension , e.g., a cable in threedimensions: The undeformed situation is implied by the zero-level sets of φ ( X ) and φ ( X ) (gray and yellow surfaces), evaluated in the domain of definition Ω X (black box).The deformed configuration results from the displacement field u ( X ) and x = X + u . The focus is on one-dimensional manifolds in R such as cables in the three-dimensionalspace. As mentioned before, two level-set functions φ and φ are needed for the geometrydefinition, Γ X = { X ∈ Ω X : φ ( X ) = 0 and φ ( X ) = 0 } , see Fig. 5. The d -dimensional displacement field u is given as before with classical deriva-tives ∇ X u ( X ) and the related deformation gradient F Ω as in Eq. (2.11).The two normal vectors associated to the deformed and undeformed configuration each,are given for i = { , } as N (cid:63)i = ∇ X φ i ( X ) for X ∈ Γ X , n (cid:63)i = ∇ x φ i ( X ( x )) = F − TΩ · N (cid:63)i for x ∈ Γ x . One may then compute the unique tangent vectors T (cid:63) = N (cid:63) × N (cid:63) , t (cid:63) = n (cid:63) × n (cid:63) . The projectors P and p follow using Eqs. (2.7) and (2.8), respectively. It is noted that8 Tangential differential calculus in finite strain theory the same projector P is obtained when using N (cid:63) and the orthogonalized normal vector N (cid:63) = N (cid:63) − N (cid:63) · N (cid:63) N (cid:63) · N (cid:63) · N (cid:63) and P = I − N ⊗ N − N ⊗ N with N = N (cid:63) (cid:107) N (cid:63) (cid:107) , N = N (cid:63) (cid:107) N (cid:63) (cid:107) , analogously for p . As before, these projectors are used to determine surface gradients ofscalar and vector functions as in Eqs. (2.12) and (2.13). The line stretch is given as Λ = (cid:107) t (cid:63) (cid:107)(cid:107) T (cid:63) (cid:107) · det F Ω = (cid:107) n (cid:63) × n (cid:63) (cid:107)(cid:107) N (cid:63) × N (cid:63) (cid:107) · det F Ω . In this section, purely based on geometric considerations related to the undeformed anddeformed situation, a number of useful quantities and operators are given following theconcept of the TDC. The situation is summarized in Table 1 for parametric manifolds andin Table 2 for implicit ones. For the mathematical equivalence of these two descriptionsand more details, see, e.g., [15]. The order of the rows in the tables is determined by theinformation available for parametric and implicit manifolds. Thereby, it is made sure thatthe (geometric and differential) quantities of interest may be computed in this order.It was found above that for parametrized manifolds, tangent vectors result naturally asprimary quantities through the existence of Jacobi matrices. For problems with codi-mension , one may then obtain normal vectors as secondary quantities (e.g., by a crossproduct of tangent vectors), however, for higher codimensions, no unique normal vectorexists. For implicit manifolds, the situation is rather the opposite: Here, normal vectorsresult naturally through the gradients of level-set functions. A unique tangent vector isonly applicable for one-dimensional manifolds and may, for example, be computed by across product of the normal vectors. This situation is an example of the duality in theparametric and implicit description of manifolds. Based on the previous definitions, some additional differential operators are introducedwhich apply to parametric and implicit manifolds equivalently. .5 Further definitions R d with d = { , } (one-dimensional manifolds) membranes in R (two-dimensional manifolds)undeformed config. Γ X X ( r ) : Ω r ⊂ R → Γ X ⊂ R d X ( r ) : Ω r ⊂ R → Γ X ⊂ R Jacobi matrix w.r.t. X ( r ) and auxiliary operators J ( r ) = ∇ r X ( r ) , G = J T · J , Q = J · G − surface gradientsw.r.t. Γ X ∇ Γ X f ( r ) = Q · ∇ r f ( r ) , ∇ Γ , dir X u = ∇ r u · Q T surface deformationgradient F Γ F Γ = ∇ Γ , dir X x ( X ) = I + ∇ Γ , dir X u deformed config. Γ x x ( X ) = X + u ( X ) ⇔ x ( r ) = X ( r ) + u ( r ) Jacobi matrix w.r.t. x ( r ) and help operators j ( r ) = ∇ r x ( r ) = F Γ · J , g = j T · j , q = j · g − surface gradientsw.r.t. Γ x ∇ Γ x f ( r ) = q · ∇ r f ( r ) , ∇ Γ , dir x u = ∇ r u · q T tangent vector(s)in undef. config. Γ X T (cid:63) = ∂ X ∂r , T = T (cid:63) (cid:107) T (cid:63) (cid:107) T (cid:63) = ∂ X ∂r , T (cid:63) = ∂ X ∂s T (cid:63) = T (cid:63) − T (cid:63) · T (cid:63) T (cid:63) · T (cid:63) · T (cid:63) T i = T (cid:63)i (cid:107) T (cid:63)i (cid:107) with i = { , , } tangent vector(s)in def. config. Γ x t (cid:63) = F Γ · T (cid:63) , t = t (cid:63) (cid:107) t (cid:63) (cid:107) t (cid:63) = F Γ · T (cid:63) , t (cid:63) = F Γ · T (cid:63) t (cid:63) = t (cid:63) − t (cid:63) · t (cid:63) t (cid:63) · t (cid:63) · t (cid:63) t i = t (cid:63)i (cid:107) t (cid:63)i (cid:107) with i = { , , } projectors P = T ⊗ T p = t ⊗ t P = T ⊗ T + T ⊗ T p = t ⊗ t + t ⊗ t line/area stretch Λ = √ det g √ det G = (cid:107) t (cid:63) (cid:107)(cid:107) T (cid:63) (cid:107) Λ = √ det g √ det G = (cid:107) t (cid:63) × t (cid:63) (cid:107)(cid:107) T (cid:63) × T (cid:63) (cid:107) relation ∇ Γ x f = W · ∇ Γ X f W = q (cid:0) Q T · Q (cid:1) − Q T Table 1: Geometric quantities and differential operators for parametric manifolds. Onlytangent vectors are considered here although normal vectors may be computed for mani-folds with codimension .0 Tangential differential calculus in finite strain theory cables in R , membr. in R (manifolds with codim. 1) cables in R (manifolds with codim. 2)undeformed config. Γ X Γ X = { X ∈ Ω X : φ ( X ) = 0 } Γ X = { X ∈ Ω X : φ ( X ) = 0) and φ ( X ) = 0 } deformed config. Γ x Γ x = { x ∈ Ω x : φ ( X ( x )) = 0 } Γ x = { x ∈ Ω x : φ ( X ( x )) = 0) and φ ( X ( x )) = 0 } classical deformationgradient F Ω F Ω = ∇ X x ( X ) = I + ∇ X u ( X ) normal vector(s)in undef. config. Γ X N (cid:63) = ∇ X φ, N = N (cid:63) (cid:107) N (cid:63) (cid:107) N (cid:63) = ∇ X φ , N (cid:63) = ∇ X φ N (cid:63) = N (cid:63) − N (cid:63) · N (cid:63) N (cid:63) · N (cid:63) · N (cid:63) N i = N (cid:63)i (cid:107) N (cid:63)i (cid:107) with i = { , , } normal vector(s)in def. config. Γ x n (cid:63) = F − TΩ · N (cid:63) , n = n (cid:63) (cid:107) n (cid:63) (cid:107) n (cid:63) = F − TΩ · N (cid:63) , n (cid:63) = F − TΩ · N (cid:63) n (cid:63) = n (cid:63) − n (cid:63) · n (cid:63) n (cid:63) · n (cid:63) · n (cid:63) n i = n (cid:63)i (cid:107) n (cid:63)i (cid:107) with i = { , , } projectors P = I − N ⊗ N p = I − n ⊗ n P = I − N ⊗ N − N ⊗ N p = I − n ⊗ n − n ⊗ n line/area stretch Λ = (cid:107) n (cid:63) (cid:107)(cid:107) N (cid:63) (cid:107) · det F Ω Λ = (cid:107) n (cid:63) × n (cid:63) (cid:107)(cid:107) N (cid:63) × N (cid:63) (cid:107) · det F Ω surface gradientsw.r.t. Γ X ∇ Γ X f = P · ∇ X f ∇ Γ , dir X u = ∇ X u · P surface deformationgradient F Γ F Γ = I + ∇ Γ , dir X u relation ∇ Γ x f = W · ∇ Γ X f W = p · F − TΩ surface gradientsw.r.t. Γ x ∇ Γ x f = p · ∇ x f = W · ∇ X f ∇ Γ , dir x u = ∇ x u · p = ∇ X u · W T Table 2: Geometric quantities and differential operators for implicit manifolds. Onlynormal vectors are considered here although tangent vectors may be computed for cables(one-dimensional manifolds) as well. .5 Further definitions The covariant surface gradient of a vector function u ( X ) : Γ X → R d is based on theprojection of the directional one (see Eqs. (2.2) and (2.13) for the parametric and im-plicit situation, respectively) onto the tangent space. With respect to the undeformedconfiguration, it is defined as ∇ Γ , cov X u ( X ) = P · ∇ Γ , dir X u ( X ) . (2.15)Concerning the surface divergence of vector functions u ( X ) and tensor functions A ( X ) :Γ X → R d × d , there holds Div Γ u ( X ) = tr (cid:16) ∇ Γ , dir X u (cid:17) = tr (cid:16) ∇ Γ , cov X u (cid:17) =: ∇ Γ X · u , (2.16) Div Γ A ( X ) = Div Γ ( A , A , A )Div Γ ( A , A , A )Div Γ ( A , A , A ) =: ∇ Γ X · A . (2.17)The divergence operator with respect to the deformed configuration follows accordingly as div Γ u ( X ) = ∇ Γ x · u and div Γ A ( X ) = ∇ Γ x · A , respectively. Unit normal and tangent vectors on manifolds have already been used before and existwith respect to the deformed and undeformed configuration, respectively. For example,the unit normal vector on an undeformed membrane is N ( X ) for all X ∈ Γ X and, afterthe deformation, n ( x ) for all x ∈ Γ x . It is important to note that, for physical reasons, themanifolds used in this work are bounded . The boundary of the undeformed configurationis labeled ∂ Γ X and in the deformed situation ∂ Γ x .There exists a co normal unit vector N ∂ Γ along the boundary ∂ Γ X which is in the tangentplane of the manifold yet normal to ∂ Γ X . This vector points out of the manifold inthe direction which naturally extends the manifold, see Fig. 6. The computation of theconormal vectors is straightforward (often using cross products) and depends on q and d . In the deformed configuration, the situation is similar for computing n ∂ Γ ( x ) for all x ∈ ∂ Γ x . In the context of the definition of boundary value problems on manifolds, theconormal vectors play a crucial role for the consideration of boundary conditions as shall2 Mechanical model and governing equations (a) cable in R (b) membrane in R Figure 6: Normal vectors, N and n , and conormal vectors, N ∂ Γ and n ∂ Γ , in undeformedand deformed manifolds. The vectors T ∂ Γ and t ∂ Γ in (b) point in tangential direction alongthe boundary.be seen in Section 3.1.5. To derive the weak form of the governing equations later on, the following divergencetheorem on manifolds is needed [12, 14], (cid:90) Γ X u · Div Γ A dΓ = − (cid:90) Γ X ∇ Γ , dir X u : A dΓ+ (cid:90) Γ X κ · u · A · N dΓ+ (cid:90) ∂ Γ X u · A · N ∂ Γ d ∂ Γ , (2.18)where ∇ Γ , dir X u : A = tr (cid:16) ∇ Γ , dir X u · A T (cid:17) is a matrix scalar product. The mean curvatureis κ = tr ( H ) with H = ∇ Γ , dir X N = ∇ Γ , cov X N being the second fundamental form. For in-plane tensor functions with A = P · A · P , the term involving the curvature κ vanishesand one finds ∇ Γ , dir X u : A = ∇ Γ , cov X u : A . In Section 2, a number of geometric quantities (such as normal vectors, projectors, area/linestretches etc.) and differential operators related to (surface) gradients are introduced. Itwas shown how these quantities are obtained for parametrized and implicitly defined man-ifolds. The focus is now turned to the mechanics and the procedure follows the classical .1 Governing equations in strong form all tensors consideredin the following refer to the parametric as well as implicit situation. These tensors havedimensions d × d (with d being the dimension into which the cable or membrane is im-mersed). A tensor A is called “in-plane” or “tangential” to the undeformed configuration Γ X if A = P · A · P and to the deformed configuration Γ x if A = p · A · p . An in-plane ( d × d ) -tensor has q non-zero eigenvalues representing the principal mechanical quantity(with q being the dimension of the manifold: q = 1 for cables, q = 2 for membranes).Starting point is the surface deformation gradient F Γ ( X ) at X ∈ Γ X , specified previouslyin Eq. (2.3). It may also be seen as a geometrical quantity mapping tangent vectors fromthe undeformed to the deformed configuration. It is also noted that the situation alsoapplies to the “volumetric” case (where q = d = 2 are flat shells in two dimensions and q = d = 3 are volumetric continua in three dimensions). In this case, F Γ = F Ω as inEq. (2.11), and for the projectors, P = p = I . In that sense, the presented mechanicaloutline below applies to cables, membranes and continua in a unified sense. Based on the surface deformation gradient, the directional and tangential Cauchy-Greenstrain tensors are defined as E dir = / · (cid:0) F TΓ · F Γ − I (cid:1) , (3.1) E tang = P · E dir · P , (3.2)respectively. The Euler-Almansi strain tensors are e dir = / · (cid:16) I − (cid:0) F Γ · F TΓ (cid:1) − (cid:17) , (3.3) e tang = p · e dir · p , (3.4)where e tang is tangential to the deformed configuration Γ x . As usual, there holds e dir = F − TΓ · E dir · F − (which is not true for the tangential versions of these strain tensors).4 Mechanical model and governing equations
Conjugated stress tensors are introduced next and only the tangential versions are consid-ered. Generally speaking, we assume some hyper-elastic material with an elastic energyfunction
Ψ ( E tang ) and obtain the second Piola-Kirchhoff stress tensor as S = ∂ Ψ ∂ E tang . Forsimplicity, only Saint Venant–Kirchhoff solids are considered herein and there follows S = λ · trace ( E tang ) · P + 2 µ E tang , (3.5) = P · ( λ · trace ( E dir ) · I + 2 µ E dir ) · P , with S being tangential to Γ X . λ and µ are the Lamé constants; for cables λ becomes .On the other hand, the Cauchy stress tensor reads σ = 1Λ · F Γ · S · F TΓ , (3.6)where Λ is a line stretch for cables and an area stretch for membranes when undergoingthe displacement, see Section 2. For the volumetric case ( q = d ), Λ = det F Ω is thevolumetric stretch. The Cauchy stress is tangential to the deformed configuration Γ x since F Γ · P = p · F Γ · P and P · F TΓ = P · F TΓ · p , hence σ = p · σ · p . The first Piola-Kirchhoffstress tensor is given by K = F Γ · S (3.7)and there holds K = K · P = p · K . For every point X ∈ Γ X and its mapped counterpart x ( X ) ∈ Γ x , we have the followingequality, S ( X ) : E tang ( X ) = ( σ ( x ) : e tang ( x )) · Λ ( X ) , (3.8)where : represents the matrix scalar product operator. In this sense the two stress tensors S and σ are conjugated to their related strain tensors E tang and e tang , respectively. It isnoted that S : E tang = S : E dir and σ : e tang = σ : e dir which will be important later. Furthermore, the result of these matrix scalar productsmay also be derived by the non-zero eigenvalues S i , E tang ,i , σ i , e tang ,i , i = 1 , . . . , q , of the .1 Governing equations in strong form S , E tang , σ , e tang , respectively. Hence, we obtain S : E tang = q (cid:88) i =1 S i · E tang ,i and σ : e tang = q (cid:88) i =1 σ i · e tang ,i . A crucial aspect of finite strain theory is that equilibrium is to be fulfilled in the deformed configuration which is expressed in strong form as div Γ σ ( x ) = − f ( x ) ∀ x ∈ Γ x , (3.9)where f are body forces. Recall from (2.17) that div Γ σ = ∇ Γ , dir x · σ = ∇ Γ , cov x · σ is thedivergence of the Cauchy stress tensor with respect to Γ x . Furthermore, we have theidentity Div Γ K ( X ) = div Γ σ ( x ( X )) · Λ ( X ) (3.10)with Div Γ K = ∇ Γ , dir X · K = ∇ Γ , cov X · K being the divergence of the first Piola-Kirchhoffstress tensor from Eq. (3.7) with respect to Γ X . In order to transform the derivatives inthe divergence operators from the undeformed to the deformed situation, use Eqs. (2.6)and (2.14) for parametric and implicit manifolds, respectively. Due to F ( X ) = f ( x ( X )) · Λ ( X ) , the equilibrium in Γ x can be stated equivalently to Eq. (3.9) based on quantitiesin the undeformed configuration as Div Γ K ( X ) = − F ( X ) ∀ X ∈ Γ X . (3.11) The domain of interest is a bounded manifold where the boundary ∂ Γ falls into the two non-overlapping parts ∂ Γ D and ∂ Γ N , which holds in the deformed and undeformed configuration Γ X and Γ x , respectively. Hence, the boundary conditions in the deformed configurationare u ( x ) = ˆ g ( x ) on ∂ Γ x , D , (3.12) σ ( x ) · n ∂ Γ ( x ) = ˆ h ( x ) on ∂ Γ x , N , (3.13)6 Mechanical model and governing equations where ˆ g are prescribed displacements and ˆ h are tractions (force per area for q = 3 , force perlength for q = 2 or a single force for q = 1 ). Note that for ropes and membranes, ˆ h mustbe in the tangent space of the deformed manifold in order to satisfy the equilibrium dueto the absence of bending stresses or transverse shear stresses. The equivalent boundaryconditions formulated in the un deformed configuration are u ( X ) = ˆ G ( X ) on ∂ Γ X , D , (3.14) K ( X ) · N ∂ Γ ( X ) = ˆ H ( X ) on ∂ Γ X , N , (3.15)where ˆ G and ˆ H have similar interpretations as before. The relation between ˆ h and ˆ H is ˆ H ( X ) = ¯Λ( X ) · ˆ h ( x ) , where ¯Λ = for q = 1 , d = { , } (cables) , line stretch along the boundary for q = 2 , d = 2 (shells) or (membranes) , area stretch of the face at the boundary for q = 3 , d = 3 (continuum) . Further information about boundary conditions for ropes and membranes are given in [8].With the boundary conditions above, the complete second-order boundary value problem(BVP) is defined in the deformed and undeformed configuration. The obtained BVP inthe frame of the TDC is valid for explicitly and implicitly defined manifolds and does notrely on curvilinear coordinates implied by a parametrization, which are typically used inclassical approaches, see, e.g., [8, 10]. Therefore, the proposed formulation of ropes andmembranes based on the TDC is more general compared to the classical theory.
For stating the governing equations in weak form, the following test and trial functionspaces are introduced S u = (cid:110) v ∈ (cid:2) H (Γ X ) (cid:3) d : v = ˆ G on ∂ Γ X , D (cid:111) , (3.16) V u = (cid:110) v ∈ (cid:2) H (Γ X ) (cid:3) d : v = on ∂ Γ X , D (cid:111) , (3.17) .2 Governing equations in weak form H is the Sobolev space of functions with square integrable first derivatives. Thetask is to find u ∈ S u such that for all w ∈ V u , there holds η · (cid:90) Γ X ∇ Γ , dir X w : K ( u ) dΓ = η · (cid:90) Γ X w · F dΓ + (cid:90) ∂ Γ X , N w · ˆ H d ∂ Γ . (3.18)where η = A for q = 1 , d = { , } is the cross section of the cable ,t for q = 2 , d = { , } is the thickness of the shell/membrane , for q = 3 , d = 3 , i.e., a continuum . The integrals in Eq. (3.18) are one-, two-, or three-dimensional for cables, membranes andcontinua, respectively. The multiplication with η ensures that the units of the integrationare always consistent. Hence, it is possible to naturally consider situations where cables,membranes and continua are coupled in one setup by simply adding up the correspondingintegrals as in Eq. (3.18) for each structure. In order to obtain Eq. (3.18), we appliedthe usual procedure for converting the strong form to a weak form: Multiply Eq. (3.11)with test functions, integrate over the domain Γ X , and apply the divergence theorem fromEq. (2.18). It is noteworthy that the curvature term from Eq. (2.18) vanishes also forcables and membranes due to K · N = .The weak form stated above is related to energy minimization in the sense that (cid:90) Γ X ∇ Γ , dir X w : K ( u ) dΓ = (cid:90) Γ X δ E tang ( u ) : S ( u ) dΓ , where δ is the variational operator. An immediate consequence of Eq. (3.8) is that one may obtain the same stored potentialenergy of the deformed body by integrating over the undeformed or deformed configurationas follows e ( u ) = 12 η · (cid:90) Γ x e tang ( u ) : σ ( u ) dΓ , (3.19) = 12 η · (cid:90) Γ X E tang ( u ) : S ( u ) dΓ . (3.20)8 Discretization and numerical methods (a) cable in R (b) cable in R (c) membrane in R Figure 7: The situation in the Surface FEM for (a) cables in R , (b) cables in R , and (c)membranes in R . The discretized domains are shown for the undeformed and deformedsituations, Γ h X and Γ h x , respectively. In order to solve the boundary value problem, i.e., to approximate the sought displace-ments, one may use two fundamentally different approaches. Possibly the more intuitiveone is to discretize the cable(s) or membrane(s) by curved ( q -dimensional) line or surfaceelements, respectively. Then, each element results from an (often isoparametric) map ofsome reference element so that this approach is naturally linked to the parametric de-scription of manifolds as discussed in Section 2.2. This is the classical approach labeledSurface FEM herein. An alternative is to use a ( d -dimensional) background mesh for theapproximation of the weak form. That is, higher-dimensional shape functions (than thedimension of the manifold) are used and evaluated on the trace of the manifold only. Thisapproach is called Trace FEM or Cut FEM. It is naturally related to an implicit manifolddescription as discussed in Section 2.3.It is important to note that the classical definition of finite strain theory based on curvi-linear coordinates does not cover the latter approach. This is another reason why thepresented formulation based on the TDC is more general as it supports both, the Surfaceand Trace FEM. For the discussion below, we assume the manifold case, hence, q < d .With q = d , the situation results in the standard FEM which is not further outlined here. .1 Surface FEM Starting point is the discretization Γ h X of the undeformed cable ( q = 1 ) or membrane ( q = 2 )by a line or surface mesh, respectively. Herein, we use higher-order q -dimensional Lagrangeelements with equally spaced nodes in the reference element. The nodal coordinates in theundeformed configuration are labeled X i with i = 1 , . . . , n q and n q being the number ofnodes in the mesh, see Fig. 7. The resulting shape functions M qi ( X ) span a C -continuousfinite element space as Q h Γ X := (cid:40) v h ∈ C (Γ h X ) : v h = n q (cid:88) i =1 M qi ( X ) · ˆ v i with ˆ v i ∈ R (cid:41) ⊂ H (Γ h X ) . (4.1) M qi ( X ) are obtained by isoparametric mappings from the q -dimensional reference elementto the physical elements in d dimensions. Based on Eq. (4.1), the following discrete testand trial function spaces are introduced S h Γ X = (cid:110) v h ∈ (cid:2) Q h Γ X (cid:3) d : v h = ˆ G on ∂ Γ h X , D (cid:111) , (4.2) V h Γ X = (cid:110) v h ∈ (cid:2) Q h Γ X (cid:3) d : v h = on ∂ Γ h X , D (cid:111) . (4.3)The discrete weak form of Eq. (3.18) reads as follows: Given Lamé constants ( λ, µ ) ∈ R + ,body forces F ∈ R d on Γ h X , tractions ˆ H ∈ R d on ∂ Γ h X , N , find the displacement field u h ∈ S h Γ X such that for all test functions w h ∈ V h Γ X there holds in Γ h X η · (cid:90) Γ h X ∇ Γ , dir X w h : K ( u h ) dΓ = η · (cid:90) Γ h X w h · F dΓ + (cid:90) ∂ Γ h X , N w h · ˆ H d ∂ Γ . (4.4)The sought discrete displacement field u h ( X ) is obtained solving a non-linear system ofequations for the n DOF = d · n q nodal values (degrees of freedom) as usual in the contextof finite strain theory. Let there be a d -dimensional background mesh into which the manifold is completely im-mersed. Only those elements and corresponding nodes are considered that are cut by themanifold, see Fig. 8. They may be labeled “active” elements and nodes, all others areneglected. The shape functions of the active nodes are constructed by (often isoparamet-0 Discretization and numerical methods (a) cable in R (b) cable in R (c) membrane in R Figure 8: The situation in the Trace FEM for (a) cables in R , (b) cables in R , and (c)membranes in R . The discretized domains are shown for the undeformed and deformedsituations, Ω h X and Ω h x , respectively. Only the black background elements and their nodesare active . The undeformed and deformed manifolds coincide with those shown in Fig. 7. .2 Trace FEM (a) active background elements (b) integration cells Figure 9: (a) Active (black) elements in a background mesh are those cut by the manifold.(b) Integration points have to be identified within the active background elements. There,the shape functions of the background elements are evaluated.ric) mappings from a d -dimensional reference element, but the shape functions are onlyevaluated on the q -dimensional manifold.The Trace FEM is a fictitious domain method (FDM) for PDEs on manifolds [33, 32,25, 23]. As in any FDM, there is no boundary-conforming mesh but a background mesh,herein further complicated by the fact that background mesh and manifold have differentdimensions. In general, the following issues have to be properly addressed:1. Integration points have to be defined for the integration of the weak form of thegoverning equations—only at these points, shape functions are evaluated. This re-quires the identification of the zero-level set of some level-set function φ ( X ) withineach active background element, see Fig. 9. The situation may be further compli-cated, when the boundary of the manifold is within the background element, whichmay be defined by additional slave level-set functions ψ i ( X ) as mentioned in Sec-tion 2.3. The placement of integration points is an important and challenging task,in particular with higher-order accuracy. For an overview, we refer to the references[18, 34, 19, 21]. It is useful to evaluate given level-set functions at the active nodes andinterpolate them in-between. The identification of the zero-level sets and placementof integration points may then be achieved in the d -dimensional reference elementwhich simplifies the evaluation of shape functions in the background elements. Ofcourse, the interpolated zero-level set is only a (higher-order) approximation of the2 Discretization and numerical methods exact manifolds Γ X and Γ x and labeled Γ h X and Γ h x , respectively. In the TraceFEM context, Γ h X and Γ h x may be seen as integration cells, see Fig. 9(b), to defineintegration points, they do not imply shape functions.2. The treatment of boundary conditions is a challenging task in FDMs as it is im-possible to directly prescribe values of the nodes in the active background elements.The additional constraints may, in principle, be enforced using penalty methods, La-grange multiplier methods, or Nitsche’s method. The latter has been developed tobe a standard choice in FDMs because the equations are formulated in a consistentway without needing additional degrees of freedom [31]. In the standard (symmetric)form of the Nitsche’s method, stabilization parameters are required. In the simplestcase, these parameters may be set to a fixed user-defined number [6, 26], however,for background elements which are cut by a tiny fraction of the manifold, resulting insmall supports of the shape functions, this may lead to unsatisfactory results. An al-ternative is to compute the stabilization parameters based on global or element-wisegeneralized eigenvalue problems [16, 40]. However, for the awkward cut situations,this may result in unbounded values resulting in similar problems known for penaltymethods [11]. Therefore, we prefer the non -symmetric Nitsche’s method herein withthe main advantage that an additional stabilization is not required for imposingboundary conditions [3, 41].3. Stabilization is still necessary when applying FDMs in the context of PDEs on man-ifolds to ensure the regularity of the resulting system of equations. This may betraced back to two sources: One is found in the shape functions with small supportsand the other in the fact that the approximated displacement field on the manifoldmay not necessarily be represented by a unique set of nodal values in the backgroundmesh. That is, the background shape functions restricted to the manifold build aframe but not necessarily a basis [23, 38, 32]. Fortunately, different stabilizationapproaches exist to cure both issues and we refer to the overview given in [32] forthe Trace FEM. Herein, we use the “normal derivative volume stabilization”, intro-duced for scalar-valued problems in [24, 7] and in [25] for vector-valued problems.This stabilization technique enables higher-order accurate results, does not changethe sparsity pattern of the stiffness matrix, and only first derivatives are needed.With these comments made, we are ready to define the discrete weak form for the TraceFEM. Let Ω h X be the background mesh into which the undeformed configuration Γ X is .2 Trace FEM active elements and nodes are present in Ω h X . The resulting shape functions M di ( X ) span a C -continuous finite element space on the manifold as Q h Ω X := (cid:40) v h ∈ C (Ω h X ) : v h = n d (cid:88) i =1 M di ( X ) · ˆ v i with ˆ v i ∈ R (cid:41) ⊂ H (Ω h X ) . (4.5)Based on Eq. (4.5), the following discrete trace test and trial function spaces are introduced T h Γ X = (cid:110) v h ∈ (cid:2) Q h Ω X (cid:3) d : v h only on Γ h X (cid:111) , (4.6) U h Γ X = T h Γ X . (4.7)For the Trace FEM, the discrete weak form of Eq. (3.18) is: Given Lamé constants ( λ, µ ) ∈ R + , body forces F ∈ R d on Γ h X , tractions ˆ H ∈ R d on ∂ Γ h X , N , and stabilization parameter ρ , find the displacement field u h ∈ T h Γ X such that for all test functions w h ∈ U h Γ X thereholds in Γ h X η · (cid:90) Γ h X ∇ Γ , dir X w h : K ( u h ) dΓ − (cid:90) ∂ Γ h X , D w h · [ K ( u h ) · N ∂ Γ X ] d ∂ Γ (cid:124) (cid:123)(cid:122) (cid:125) boundary term due to w h (cid:54) = on ∂ Γ h X , D + (4.8) (cid:90) ∂ Γ h X , D (cid:16) u h − ˆ G (cid:17) · [ K ( w h ) · N ∂ Γ X ] d ∂ Γ (cid:124) (cid:123)(cid:122) (cid:125) Nitsche term + ρ (cid:90) Ω h X ( N e · ∇ X u h ) · ( N e · ∇ X w h ) dΩ (cid:124) (cid:123)(cid:122) (cid:125) stabilization term = η · (cid:90) Γ h X w h · F dΓ + (cid:90) ∂ Γ h X , N w h · ˆ H d ∂ Γ . In comparison to Eq. (4.4), additional terms occur in the discrete weak form due to the weakenforcement of essential boundary conditions with Nitsche’s method and the stabilization.The sought discrete displacement field u h ( X ) is obtained solving a non-linear system ofequations for the n DOF = d · n d nodal values (degrees of freedom).Note that the stabilization term is the only term which is not evaluated on the mani-fold Γ h X but in the volumetric background mesh Ω h X (using standard Gauss integration).Therefore, one has to extend the normal vector N ( X ) from the undeformed manifold Γ h X to the neighborhood, resulting in N e ( X ) for all X ∈ Ω h X . This is particularly simple forimplicitly defined manifolds, e.g., using N e ( X ) = ∇ X φ h ( X ) (cid:107)∇ X φ h ( X ) (cid:107) for all X ∈ Ω h X . Further-more, in the stabilization term, the classical gradient operator ∇ X is used instead of the4 Numerical results surface operators used in the other terms. In [24], it is recommended that the stabilizationparameter should be chosen in the range O ( h ) (cid:46) ρ (cid:46) O ( h − ) , where h is the element sizein the active background mesh.A remark is added concerning slip supports because the above mentioned weak form ratherexpects that all displacement components are prescribed through ˆ G along the Dirichletboundary ∂ Γ h X , D . In the Nitsche’s method, displacement constraints in selected, arbitraryunit directions v d with magnitude ˆ G may be prescribed by replacing the correspondingterms in Eq. (4.8) with − (cid:90) ∂ Γ h X , D ( w h · v d ) [ K ( u h ) · N ∂ Γ X ] · v d d ∂ Γ+ (cid:90) ∂ Γ h X , D (cid:16) u h · v d − ˆ G (cid:17) [ K ( w h ) · N ∂ Γ X ] · v d d ∂ Γ . (4.9) A number of test cases for ropes and membranes in two and three dimensions are consideredin this section. The numerical results focus on the convergence rates for two different typesof errors. The “energy error” ε e compares the approximated stored elastic energy with theanalytical one, ε e = | e ( u ) − e ( u h ) | , (5.1)with e computed based on Eq. (3.19). The analytical energy e ( u ) may also be computed byan overkill approximation, i.e., based on an extremely fine mesh with higher-order elements.Provided that geometry and boundary conditions allow for sufficiently smooth solutions,the expected convergence rates in this error norm are p + 1 with p being the order of theelements.The “residual error” ε res integrates the error in the equilibrium as stated in Eq. (3.9), thatis, ε res = (cid:118)(cid:117)(cid:117)(cid:116) n el (cid:88) e=1 (cid:90) Γ h, e X r ( u h ) · r ( u h ) dΓ with r ( u h ) = div Γ σ ( u h ) + f ( x ) (5.2)This error obviously vanishes for the analytical solution. It is important to note that theintegrand in (5.2) involves second-order derivatives. Therefore, the integral must not becarried out over the whole (discretized) domain Γ h X but integrated element by element .1 Membrane with given deformation C -continuous shape functions feature jumps, are neglected in computing ε res . Due to the presence of second-order derivatives, the expected convergence rates are p − which also indicates that higher-order elements are crucial for convergence in ε res . In the first test case, a membrane in the shape of a half sphere with radius r = 1 . undergoes a prescribed displacement and the stored elastic energy is computed from theviewpoint of the Surface FEM and the Trace FEM. That is, in the Surface FEM, surface meshes with different resolutions and element orders are generated. See Fig. 7(c) for someexample mesh composed by quadratic elements. The displacements u ( X ) = / + / ( X + 1) − / − / [1 − ( X + Y )] − / (5.3)are evaluated at the nodes and interpolated based on the shape functions implied by the surface meshes, yielding u h ( X ) ; see Fig. 7(c) for the resulting deformed membrane. Then,the elastic energy of the deformed configuration e ( u h ) is computed with Eq. (3.19).For the Trace FEM viewpoint, background meshes of different resolutions and orders aregenerated in Ω X = [ − , × [ − , × [0 , and the geometry is defined based on the level-set function φ ( X ) = (cid:107) X (cid:107) − r . See Fig. 8(c) for a sketch of the situation using quadraticbackground elements. Then, the (active) nodes of the background mesh are deformed bythe given displacement field (5.3), yielding u h ( X ) based on the shape functions impliedby the background meshes. This displacement field living in the whole background meshis only evaluated on the membrane surface in order to compute the stored energy e ( u h ) according to the Trace FEM.We set the Lamé constants to λ = 3 and µ = 2 . The resulting energy is given by the value e ( u ) = 1 . . In Fig. 10, the convergence results for the various meshes areshown for the Surface and the Trace FEM. It is seen that in both cases optimal convergenceresults are achieved. The energy error converges one order higher than expected for evenelement orders. In the Trace FEM, the convergence curves are less smooth than in theSurface FEM because the approximation spaces are not nested upon refinement; this iswell-known for results obtained with FDMs in general.6 Numerical results -12 -10 -8 -6 -4 -2 linearquadraticcubic4th-order5th-order6th-order (a) Surface FEM, ε e -10 -8 -6 -4 -2 linearquadraticcubic4th-order5th-order6th-order (b) Trace FEM, ε e Figure 10: Convergence results for test case 1: The energy error ε e for the (a) Surface FEMand (b) Trace FEM.It is thus seen that the Surface FEM as well as the Trace FEM have the potential toachieve optimal results. For all other test cases below, we shall now obtain the discretedisplacement fields based on solving the non-linear systems of equations resulting from theweak forms given in Section 4. The second test case considers a rope in two dimensions as shown in Fig. 11. The crosssection of the rope is A = 0 . , Young’s modulus is E = 10 000 , and the Lamé constantsare µ = / E and λ = 0 . The left support is located at (0 , / ) and the right at (1 , . Thegeometry may be given in parametric form for r ∈ (0 , as X ( r ) = (cid:34) X ( r ) Y ( r ) (cid:35) = (cid:34) r / (1 − r ) − / sin ( π · r ) (cid:35) . Alternatively, the same geometry may be implied by the level-set function φ ( X ) = Y − [ / (1 − X ) − / sin ( π · X )] ∀ X ∈ (0 , . The structure is loaded by its own weight with F ( X ) = [0 , − · A ] T for all X ∈ Γ X .The deformed rope is illustrated in Fig. 11(a) where the color information indicates theprincipal Cauchy stress in the rope. The stored elastic energy is e = 0 . and .3 Membranes with Surface FEM (a) overview (b) Trace FEM Figure 11: (a) Sketch of test case 2 including the deformed and undeformed configuration.The colored deformed configuration shows the principal Cauchy stress. (b) The situationin the Trace FEM using two-dimensional cubic background elements.the length of the rope is increased by a factor of . due to the deformation.The displacements are computed with the Surface and Trace FEM, respectively. For theTrace FEM, the stabilization parameter in Eq. (4.8) is set to ρ = / h . The mesh res-olutions and orders are systematically varied and convergence results are seen in Fig. 12.Figs. 12(a) and (b) show the error in the elastic energy ε e and Figs. 12(c) and (d) the resid-ual error ε res . Of course, for some given element length h , the use of (curved) line meshesin the Surface FEM results in considerably less degrees of freedom than using backgroundmeshes in the Trace FEM. Therefore, the convergence studies for the Surface FEM arerealized for up to elements whereas the background meshes in the Trace FEM featureup to × elements (of which only those cut by the rope are active and, hence, takeninto account for the simulation). It is apparent from the convergence results in Fig. 12that optimal higher-order convergence rates are achieved. It is noteworthy that the SurfaceFEM with even element orders converges one order higher than expected in ε e , whereasthis is not the case for the Trace FEM. This can already be traced back to the accuracyin the numerical integration: Simply integrating the length of the rope using the Surfaceor Trace FEM viewpoint shows that an extra order in the accuracy is achieved with theSurface FEM for even element orders. For the convergence results in ε res , results for linearmeshes are omitted because second order derivatives are needed for this error measure, seeEq. (5.2). In this error norm, Surface and Trace FEM converge optimally with p − .8 Numerical results -3 -2 -1 -12 -10 -8 -6 -4 -2 linearquadraticcubic4th-order5th-order6th-order (a) Surface FEM, ε e -2 -1 -8 -6 -4 -2 linearquadraticcubic4th-order5th-order6th-order (b) Trace FEM, ε e -3 -2 -1 -6 -4 -2 quadraticcubic4th-order5th-order6th-order (c) Surface FEM, ε res -2 -1 -4 -2 quadraticcubic4th-order5th-order6th-order (d) Trace FEM, ε res Figure 12: Convergence results for test case 2: (a) and (b) show the energy error ε e , (c)and (d) the residual error ε res for the Surface and Trace FEM, respectively. .3 Membranes with Surface FEM (a) Map A, undeformed conf. (b) Map A, deformed conf. (c) Map A, displ. w (d) Map B, undeformed conf. (e) Map B, deformed conf. (f) Map B, displ. w Figure 13: Sketch of test case 3 for the two different maps A and B: (a) and (d) show theundeformed configurations Γ h X with example meshes composed by quadratic elements, (b)and (e) the deformed configurations Γ h x with von-Mises stresses, (c) and (f) top views ofthe vertical displacement fields w h . White lines show element edges in Γ h X , black lines in Γ h x .0 Numerical results
For the next test case, the deformation of membranes is approximated with the SurfaceFEM. Two different undeformed configurations are considered which result from a map ofa unit circle for map A and of a unit square for map B. The undeformed configurations Γ X are given by the parametrizations X ( r ) = / · rsc · sin ( r · s ) with √ r + s ∈ (0 , for map A, r, s ∈ ( − , for map B, (5.4)where c ∈ R is a scaling parameter in vertical direction. The resulting configurationsfor map A and B are illustrated in Figs. 13(a) and (d) for c = 0 . , respectively. Animportant difference is that the boundary is smooth for map A but involves corners formap B, later resulting in different convergence behaviors. The thickness of the membrane is t = 0 . , Young’s modulus is E = 1 000 and Poisson ratio ν = 0 . , which is easily convertedinto the Lamé parameters. The loading is gravity acting on the membrane surface with F ( X ) = [0 , , − · t ] T for all X ∈ Γ X . The whole boundary is treated as a Dirichletboundary with prescribed zero-displacements. The deformed configurations are displayedin Figs. 13(b) and (e) with computed von-Mises stresses based on the Cauchy stress tensor.The vertical displacement field is given in Fig. 13(c) and (f) for the two maps in top view,respectively.Convergence results are given in Fig. 14. For map A, where the boundary is smooth,optimal convergence rates are found in the energy error ε e and residual error ε res . For mapB, where corners are present in the membrane geometry, it is seen that the convergencerates are bounded. Only linear and quadratic elements converge optimally in ε e , higherorders improve the error level, however, not the convergence rates. It is thus confirmedthat corners in membranes have the potential to reduce the convergence rates. It was mentioned several times that the proposed framework allows for a unified treatmentof ropes and membranes (and even continua). This shall be confirmed with this last testcase where cables and membranes are coupled. The situation may be described as aninflated ball with embedded reinforcement cables, see Fig. 15(a). The original radius of .4 Coupled ropes and membranes -10 -8 -6 -4 -2 linearquadraticcubic4th-order5th-order6th-order (a) map A, ε e -4 -2 quadraticcubic4th-order5th-order6th-order (b) map A, ε res -6 -4 -2 linearquadraticcubic4th-order5th-order6th-order (c) map B, ε e quadraticcubic4th-order5th-order6th-order (d) map B, ε res Figure 14: Convergence results for test case 3: (a) and (b) show the results for the energyerror ε e and residual error ε res for map A, (c) and (d) for map B. (a) inflated ball (b) domains Γ X and Γ x (c) von-Mises stress Figure 15: Sketch of test case 4: (a) The full inflated ball reinforced with cables (blacklines), (b) deformed and undeformed domains in the simulations, (c) von-Mises stresses inthe membrane.2
Numerical results -10 -8 -6 -4 -2 linearquadraticcubic4th-order5th-order6th-order (a) energy error ε e -2 quadraticcubic4th-order5th-order6th-order (b) residual error ε res Figure 16: Convergence results for test case 4: (a) the energy error ε e and (b) the residualerror ε res .the ball is r = 1 . and the inner pressure is p = 20 . This pressure converts to a loadingof f ( x ) = p · n ( x ) which depends on the normal vector of the deformed configuration,hence on the displacements u , thereby adding a new source of non-linearity which has tobe properly considered in the Newton-Raphson loop. The membrane is defined by theparameters t = 0 . , E = 1 000 and ν = 0 . . The cables which are on the gray planes inFig. 15(a) feature a Young’s modulus of E = 1 000 000 and a cross section of A = 0 . .All other cables have a Young’s modulus of E = 500 000 . For the modeling, only oneeighth of the initial sphere is considered, see Fig. 15(b) for the deformed and undeformedsituations. Symmetry boundary conditions are applied. The resulting von-Mises stressesin the membrane are shown in Fig. 15(c). The elastic energy stored in the membrane and ropes is given as e ( u ) = 2 . .The convergence results are seen in Fig. 16 for the energy error ε e and residual error ε res . Itis again seen that the convergence rates in ε e are optimal for linear and quadratic elements.Higher orders achieve better results, however, do not improve the convergence rates. Thereason for the bounded convergence rates is found in the stress concentrations where theembedded cables meet, see Fig. 15(c).3 The modeling of ropes and membranes leads to partial differential equations on manifolds.Herein, a framework is proposed which unifies the mechanical modeling of ropes and mem-branes undergoing large displacements and, furthermore, applies to parametric and implicitdefinitions of manifolds. The fundamental ingredient is the use of Tangential DifferentialCalculus (TDC) for the definition of geometric and differential quantities on the mani-folds. The proposed TDC-based formulation is more general than the classical parametricformulation based on curvilinear coordinates because it allows for two different numericalapproaches, the Surface and the Trace FEM. The classical approach is restricted to theSurface FEM and cannot handle manifolds which are defined implicitly (unless discretizedby a surface mesh).Even in the classical setting using the Surface FEM with parametric formulations, thenew TDC-based formulation leads to a significantly different implementation, however,ultimately achieving the same results (as expected). The advantage of the TDC-basedformulation is that it also applies immediately to implicit definitions and the Trace FEM.More technically speaking, we have one element integration routine which applies to ropesand membranes (and even continua) no matter whether we are using the Surface FEM orthe Trace FEM. Of course, in the Trace FEM, additional terms have to be considered forthe stabilization.The numerical results consider various test cases with ropes and membranes in two andthree dimensions. The Surface and Trace FEM with higher-order elements are used suc-cessfully, achieving higher-order convergence rates. It is confirmed that the smoothness ofthe geometry and the coupling of ropes and membranes has an important influence on theconvergence behavior.
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