Time Integration in the Context of Energy Control and ... - CiteSeerX

Vol. 7, 3, 299-332 (2000)

Archives of Computational Methods in Engineering State of the art reviews

Time Integration in the Context of Energy Control and Locking Free Finite Elements D. Kuhl Institute for Structural Mechanics, Ruhr-University Bochum 44780 Bochum, Germany [email protected]

E. Ramm Institute of Structural Mechanics, University Stuttgart 70550 Stuttgart, Germany [email protected]

Summary In the present paper two main research areas of computational mechanics, namely the finite element development and the design of time integration algorithms are reviewed and discussed with a special emphasis on their combination. The finite element techniques are designed to prevent locking and the time integration schemes to guarantee numerical stability in non-linear elastodynamics. If classical finite element techniques are used, their combination with time integration schemes allow to avoid any modifications on the element or algorithmic level. It is pointed out, that on the other hand Assumed Stress and Enhanced Assumed Strain elements have to be modified if they are combined with energy conserving or decaying time integration schemes, especially the Energy-Momentum Method in its original and generalized form. The paper focusses on the necessary algorithmic formulation of Enhanced Assumed Strain elements which will be developed by the reformulation of the Generalized Energy-Momentum Method based on a classical one-field functional, the extension to a modified Hu-Washizu three-field functional including enhanced strains and a suitable time discretization of the additional strain terms. The proposed method is applied to non-linear shell dynamics using a shell element which allows for shear deformation and thickness change, and in which the Enhanced Assumed Strain Concept is introduced to avoid artificial thickness locking. Selected examples illustrate the locking free and numerically stable analysis.

1 INTRODUCTION In the last three decades finite element techniques and solution algorithms for dynamic structural analysis have been lively discussed in computational mechanics literature. The objective of most finite element developments is the design of reliable, efficient and locking free elements. On the other side, the development of integration algorithms is focused on accurate, numerically stable and numerically dissipative time marching schemes. However, the aspect of combining both, special finite element techniques and time stepping algorithms, such that the positive properties of both ingredients are preserved, got less attention in the literature. In order to point out interactions and necessary adjustments of finite element formulations and time integration algorithms both areas will be reviewed, classified and finally discussed in the context of their potential combination. Although already extensive the list of cited literature is by far not complete. 1.1 Finite Element Techniques Preventing Locking Locking of certain standard low order displacement based finite elements comes along in different ways, namely as volumetric locking if incompressible or nearly incompressible materials are used or as membrane, shear and thickness locking if the stress and strain space is not compatible due to the spatial discretization. Methods to remedy locking defects may be classified as follows: c 2000 by CIMNE, Barcelona (Spain).

ISSN: 1134–3060

Received: February 2000

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• Assumed stress elements by Pian [47], Pian & Chen [49] and Pian & Sumihara [50], based on the Hellinger-Reissner functional to enrich the stress space. • Reduced or selected reduced integration techniques by Zienkiewicz et al. [79], to clear the balance between the different stress spaces by a modified numerical integration. • Assumed Natural Strain elements by Hughes & Tezduyar [34], Dvorkin & Bathe [24] and Bathe & Dvorkin [6] with a variational basis of the Hu-Washizu functional given by Simo & Hughes [64], to adapt the strain field so that no locking occurs. • Incompatible displacement models by Wilson et al. [72] and Taylor et al. [71], designed by the extension of the trial functions through additional incompatible modes. • Enhanced Assumed Strain elements by Simo & Rifai [65], Simo & Armero [61] and Simo et al. [62] based on the Hu-Washizu functional and the extension of the strain tensor or the material deformation gradient by additional terms. It should also be mentioned, that a natural strategy to remedy locking defects is the design of • higher order finite elements as proposed for example by Bas¸ar [4] and Sansour [56, 57] for higher order shell theories with quadratic transverse displacement distribution across the thickness and by Rank et ´ & Sahrmann [70] and Schwab [60] for p-method plate and shell elements. al. [53], Szabo However, higher order methods require a more complex element development in the first case and a substantial increase of the number of degrees of freedom on the element level in the second case. Furthermore, low order elements may be advantageous if the problem is not smooth. 1.2 Time Integration Schemes with Energy Control Traditionally the design of implicit time integration schemes is equally controlled by the arguments of accuracy, stability and high frequency dissipation. The situation changes in non-linear dynamics because unconditionally stable time integration schemes of linear dynamics are not automatically stable in the non-linear regime. Consequently, numerical stability or its related necessary condition, namely the conservation or decay of total energy, is currently the main discussed issue of time integration schemes. In principle, three different strategies designed to guarantee the conservation or decay of total energy can be distinguished: • Numerical dissipation as published by Newmark [46], Hilber et al. [32], Wood et al. [73] and Chung & Hulbert [19] including damping of high frequency structural response. • Enforced conservation of total energy, proposed by Hughes et al. [33] and Kuhl & Ramm [40], augmenting the semidiscrete equation of motion by the stability condition as additional constraint. • Algorithmic conservation of total energy originally proposed by Simo & Tarnow [66, 67] or algorithmic decay of energy including controlled high frequency structural ˝ cz [2], Crisfield et al. [22], Kuhl & response as published by Armero & Peto Crisfield [39] and Armero & Romero [3] using algorithmic stresses instead of generalized mid-point stresses.

301

Time Integration in the Context of Energy Control and Locking Free Finite Elements

finite element techniques reduced integration techniques linear strain ¯+ε ˜ ε=ε GL strain

time integration schemes numerical dissipation energy conservation S n+ 1

incompatible displacement models

¯ +E ˜ E=E

Enhanced Assumed Strain

deformation gradient ¯ +F ˜ F =F

Assumed Natural Strain assumed stress

2

algorithmic energy control

additional dissipation S n+ 1 +ξ 2

generalized dissipation S n+1−αf +ξ enforced energy conservation

unmodified modified

Figure 1. Combination of finite element techniques and time integration schemes

1.3 Combination of Time Integration Schemes and Finite Element Techniques The following discussion is assisted by the illustration given in Figure 1. Implicit time integration schemes following the first two strategies, namely the numerical dissipation and the enforced conservation of total energy as well as their combination are based on the formulation of the dynamic equilibrium at an arbitrary configuration within the time step. This means, that stresses and the variation of strains are evaluated at the same time or configuration. Consequently, finite element techniques preventing locking are not influenced by the temporal discretization method. All above discussed methods can be used in a standard unmodified manner. For example, see the combination of an Enhanced Assumed Strain shell element and classical implicit time integration schemes in their pure and energy-momentum constraint version by Kuhl & Ramm [40]. In contrast to this, time integration schemes with the property of algorithmic conservation or decay of energy, denoted as energy controlled time integration schemes, apply different formulations for stresses and strain variations in the time domain for the calculation of internal forces and the tangential stiffness matrix. In fact, the stresses at the generalized mid-point configuration are substituted by algorithmic stresses, calculated by the combination of the stresses at the beginning and the end of the time step whereas in contrast to the stresses the strain variation is evaluated at the generalized mid-point configuration. As consequence of this algorithmic formulation of stresses and strains, some of the above mentioned finite element techniques need to be modified within the framework of energy controlled time stepping schemes, in order to preserve the positive properties of both, the locking free finite element formulation and the stable time integration. It is evident, that higher order finite elements or reduced integration techniques can be used without any modification of the element technology or the time integration scheme, because pure displacement formulations are applied. However, the reduced integration tech-

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nique may lead to a loss of the energy conservation property of the time integration scheme, and dynamically activated zero energy modes may enter as a consequence. Assumed Natural Strain formulations are based on the evolution of transverse shear strain components of plate and shell elements at specific sampling points and on an interpolation within the element by related trial functions. This procedure does not affect the algorithmic formulation of the internal forces within the framework of energy controlled time integration schemes. Consequently, the algorithmic formulation of Assumed Natural Strain elements is identical to pure displacement finite elements, i.e. algorithmic stresses and generalized mid-point strain variations are obtained in the same way as in the pure displacement based energy controlled time stepping scheme. Assumed stress finite elements are based on the Hellinger-Reissner mixed variational principle with the stresses and displacements as primary variables. If those hybrid stress elements are combined with an energy conserving time integration scheme, the formulation of the algorithmic stresses and internal forces, has to be modified. However, a satisfying algorithmic formulation is still an open question. A possible answer may be found in the equivalence of assumed stress finite elements and incompatible displacement models or Enhanced Assumed Strain elements investigated by Pian [48], Yeo & Lee [77], Braess [13] and Bischoff et al. [12]. Currently, a popular method preventing locking defects are the Enhanced Assumed Strain concepts and their predecessor, the incompatible displacement models. These methods are based on the indirect or direct enrichment of the compatible strain or deformation gradient field by incompatible terms within the modified Hu-Washizu variational principle. As basis of their combination to energy conserving or decaying time integration schemes the algorithmic formulation of the enhanced strain field and its influence on the algorithmic stress field will be developed and investigated in detail in the present paper. 1.4 Enhanced Assumed Strain Concept Enhanced Assumed Strain (EAS) methods are applied to linear and non-linear material models in the small and finite strain regime. They may be classified by the type of enhancement: ¯+ε ˜, Simo & Rifai [65]. • Enhancing the small strain tensor ε = ε ¯ +F ˜ in finite strain analysis, • Enhancing the material deformation gradient F = F Simo & Armero [61]. • Enhancing the Green-Lagrange strain tensor in the geometrically non-linear case ¯ + E, ˜ Li et al. [42] and Bu ¨ chter et al. [18]. E=E The last version of the non-linear EAS concept is very efficient and leads to similar results as the original non-linear concept by Simo & Armero [61]. The following discussion of different finite elements using the EAS concept is restricted to plates, shells and shell like three dimensional elements. They are designed to avoid: • Shear locking, enhancing the transverse shear strain, Simo & Rifai [65]. • Membrane locking enhancing the in-plane normal and shear strains, Andelfinger & Ramm [1]. • Thickness locking in an extensible director shell theory by additional transverse nor¨ chter et al. [18]. mal strains, Bu

303


❇❇locking phenomenon enhancement of ... ❇❇

thickness locking

membrane locking

shear locking

¯ +E ˜ E =E

˜ =E ˜33 G3 ⊗ G3 E

˜ =E ˜αβ Gα ⊗ Gβ E

˜ =E ˜α3 Gα ⊗ G3 E

Simo & Rifai [65]

¨chter & Bu Ramm [17]

Andelfinger & Ramm [1]

Simo & Rifai [65]

[13], [28], [36], [42], [51, 52], [54], [55], [75]

[5], [7], [8], [9], [11], [15], [16], [18], [27], [31], [35]

[5], [7], [8], [11], [15], [27], [35]

¯ +F ˜ F =F

˜ = F˜ 3 g3 ⊗ G3 F 3

˜ = F˜ α g α ⊗ Gβ F β

Betsch & Stein [10]

Betsch & Stein [10]

Simo & Armero [61] [7], [30], [37, 38], [45], [62], [69], [74], [75], [76]

[7], [25], [43]

[7]

˜ = F˜ α g α ⊗ G3 F 3 Sansour & Bocko [58]

[7], [25], [26], [43], [58]

Table 1. Development and application of the Enhanced Assumed Strain concept - references

A selection of papers concerning the development of EAS methods in the context of locking prevention is classified in Table 1 with respect to the above discussed issues. In this table the enhancements of ε and E are not distinguished, because the computational realization is nearly identical. If the related finite elements should be used in an theoretically effective manner within the framework of energy controlled time integration schemes, the EAS concept has to be adapted to the algorithmic formulation. 1.5 Time Stepping Schemes with Algorithmic Conservation or Decay of Energy The pioneering design of the unconditionally stable, implicit, energy conserving time integration scheme, called Energy-Momentum Method (EMM), by Simo & Tarnow [66] has stimulated intensive research work, for example the application to different finite elements. Truss elements are investigated by Crisfield & Shi [23] and Kuhl & Crsifield [39], beam elements by Simo et al. [68], Galvanetto & Crisfield [29] and Crisfield et al. [22] and shell elements by Simo et al. [63], Simo & Tarnow [67], Brank et al. [14] and Sansour et al. [59]. A further focus is the inclusion of controllable numerical dissipation of high frequencies. In historical order, the original algorithm and its algorithmic modifications and generalizations are characterized as follows: • Design of the EMM, Simo & Tarnow [66], by modification of the mid-point rule substituting of the mid-point stress by the average of the stresses at the beginning and the end of the time step S(un+1/2 ) → S n+1/2 = 1/2(S(un+1 ) + S(un )). ˝ cz [2] and Cr• Extension of the EMM by numerical dissipation, Armero & Peto isfield et al. [22], realized by a time shift ξ modifying the algorithmic stresses S(un+1/2 ) → S n+1/2+ξ = (1/2 + ξ)S(un+1 ) + (1/2 − ξ)S(un ). • Development of the Generalized Energy-Momentum Method (GEMM) within the scope of the Generalized-α Method (Chung & Hulbert [19]), including well-known numerical damping characteristics, by Kuhl & Crisfield [39] modifying the generalized mid-point stresses S(un+1−αf ) → S n+1−αf +ξ = (1 − αf + ξ)S(un+1 ) + (αf − ξ)S(un ).

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As the GEMM includes the original EMM as well as the extended numerical dissipative version, the present algorithmic formulation of the EAS concept leads to a fairly general version within this family of time integration schemes. 1.6 Organization of the Paper The present paper is organized as follows: In the next section the GEMM, originally designed on the basis of the semidiscrete equation of motion, will be derived from a one-field variational principle. The EAS method using directly the strain tensor is discussed in section three preparing the introduction into the GEMM in section four. Next, related finite element techniques as the incompatible displacement models, the non-linear version of the EAS method as well as classical time stepping schemes are discussed in the context of a numerically stable and locking free numerical framework. Finally, the proposed method is applied to non-linear shell dynamics preventing thickness locking in extensible director shells and simultaneously allowing for a stable time integration. Selected examples demonstrate the improved bending behavior of this EAS shell element and the numerical stability of the strain enhanced time stepping scheme. 2 GENERALIZED ENERGY-MOMENTUM METHOD As a generalized type of energy controlled time integration schemes, the Generalized EnergyMomentum Method (GEMM) by Kuhl & Crisfield [39] is chosen. This algorithm includes the second order accurate energy conserving Energy-Momentum Method (EMM) by Simo & Tarnow [66] as well as a first order accurate energy decaying scheme by Armero ˝ cz [2] and Crisfield et al. [22] as special cases. Furthermore, the classical im& Peto plicit time marching schemes proposed by Newmark [46] and Wood et al. [73] are also included. In contrast to the original literature, the algorithm is formulated on the basis of a functional using variational calculus and sequential time-space discretization. The results of both alternative formulations are identical. However, the present formulation is advantageous, if the one-field functional will be replaced by the three-field Hu-Washizu functional and modified in the context of the EAS concept. 2.1 Variational Formulation The standard one-field functional Π of an elastic body B, loaded by time dependent body and surface loads, extended by the inertial force term constitute the basis of the present family of time integration algorithms. Π is given in the Langangeian description by:

Π(u) = B

W (E(u)) dV −

¨ ) ρ dV − u · (b − u

B

u · t dA

(1)

∂B

W is the quadratic stored energy function, E is the Green-Lagrange strain tensor, u ¨ are the displacement and acceleration fields, respectively, b and t are the specified and u body and traction vectors, respectively, and ρ is the density in the reference configuration. The Green-Lagrange strain tensor is given in terms of the second order identity tensor I, the material deformation gradient F as function of the position vector in the reference configuration X and the displacement vector u. E(u) =

1 T F (u) F (u) − I , 2

F (u) =

∂(X + u) = I + ∇u ∂X

(2)

305


The variation of equation (1) with respect to the displacements yields the continuous stationary condition of non-linear elastodynamics.

δΠ(u) =

δE(u) : S(u) dV −

B

¨ ) ρ dV − δu · (b − u

B

δu · t dA = 0

(3)

∂B

Herein, the derivative of the stored energy function with respect to the Green-Lagrange strains is substituted by the second Piola-Kirchhoff constitutive stress tensor S. The second derivative of the stored energy function defines the material stiffness tensor C in the sense of a Saint Venant-Kirchhoff material which can be expressed in terms of the Lam´ e constants λ and µ, the symmetric fourth order identity tensor I 4 and I. S(u) =

∂W (E(u)) = C : E(u) , ∂E(u)

C=

∂ 2 W (E(u)) = 2µ I 4 + λ I ⊗ I ∂E(u) ⊗ ∂E(u)

2.2 Temporal Discretization

(4)

T For the time discretization, the time interval of interest [0, T ] = N n=1 [tn , tn+1 ] is subdivided in constant or adaptively controlled (see Kuhl & Ramm [41]) time steps ∆t = tn+1 − tn . ¨ n are assumed to be The state variables at the beginning of the time step un , u˙ n and u known. In the framework of an implicit time integration, the state variables at the end of the ¨ n+1 are calculated through time approximations, specified later, time step un+1 , u˙ n+1 and u and equation (3) is evaluated at the time t ∈ ]tn , tn+1 ]. Here, the stationary condition is satisfied at the generalized mid-point tn+1−αf ∈ ]tn , tn+1 ] with tn+1−αf = (1−αf )tn+1 +αf tn , where the acceleration vector is evaluated at a different sampling time tn+1−αm ∈ ]tn , tn+1 ] with tn+1−αm = (1−αm )tn+1 + αm tn ; αf and αm are time integration parameters.

δΠ(un+1−αf ) = −

B

δE(un+1−αf ) : S(un+1−αf ) dV + δun+1−αf · bn+1−αf ρ dV

B

−

B

¨ n+1−αm ρ dV δun+1−αf · u δun+1−αf · tn+1−αf dA

(5)

∂B

Accelerations, velocities, displacements, body and surface loads of the generalized midpoints tn+1−αm and tn+1−αf are generated by convex functions. ¨ n+1 + αm u ¨n ¨ n+1−αm = (1−αm ) u u

un+1−αf = (1−αf ) un+1 + αf un

bn+1−αf = (1−αf ) bn+1 + αf bn

tn+1−αf = (1−αf ) tn+1 + αf tn

(6)

The temporal discretization is completed by Newmark’s [46] approximations of the state variables, γ γ γ ¨n (un+1 − un ) − ( − 1) u˙ n − ( − 1) ∆t u β∆t β 2β 1 1 1 ¨n ¨ n+1 (un+1 ) = (un+1 − un ) − ( u˙ n − − 1) u u β∆t2 β∆t 2β

u˙ n+1 (un+1 ) =

(7)

which finally constitutes the acceleration vector at the generalized mid-point in combination with equation (6). Equations (5-7) define the variational formulation of the non-linear version of the Generalized-α Method, based on the calculation of the internal forces at the

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generalized mid-point, see Kuhl & Crisfield [39]. However, this method does not satisfy the condition of numerical stability, which necessitates the control of total energy within every time step. In order to satisfy this stability condition, the idea of Simo & Tarnow [66] is generalized within the scope of the Generalized-α Method by Kuhl & Crisfield [39] and Kuhl & Ramm [41]. The crucial point in the development of this algorithm is the substitution of the second Piola-Kirchhoff stress tensor at the generalized mid-point by the algorithmic second Piola-Kirchhoff stress tensor, defined by the combination of the stress tensors at the beginning and the end of the time step. S(un+1−αf ) → S n+1−αf +ξ = (1−αf +ξ) S(un+1 ) + (αf −ξ) S(un )

(8)

The additional algorithmic modification by the parameter ξ ≥ 0 has been originally pro˝ cz [2]. They have introduced the additional parameter in order posed by Armero & Peto to introduce numerical dissipation to the EMM applied to frictionless dynamic contact problems on the level of algorithmic contact forces. Crisfield et al. [22] have transferred this idea to the modification of the algorithmic stress tensor within the EMM and applied it to the non-linear elastodynamics of beams. Combining equations (5-8) yields the time discrete stationary condition of the GEMM.

δΠ(un+1−αf ) =

δE(un+1−αf ) : [(1−αf +ξ) S(un+1 ) + (αf −ξ) S(un )] dV

B

+

δun+1−αf B

−

1−αm 1−αm −2β 1−αm ¨ n ρ dV · (un+1 − un ) − u˙ n − u 2 β∆t β∆t 2β

δun+1−αf · bn+1−αf ρ dV −

B

δun+1−αf · tn+1−αf dA = 0

(9)

∂B

The set of time integration parameters αm , αf , β and γ reduces to one time integration parameter ρ∞ if optimized high and low frequency dissipation and second order accuracy are enforced as proposed by Chung & Hulbert [19]. αm =

2 ρ∞ − 1 , ρ∞ + 1

αf =

ρ∞ , ρ∞ + 1

β=

1 (1 − αm + αf )2 , 4

γ=

1 − αm + αf 2

(10)

In linear dynamics, the time integration parameter ρ∞ is defined as the spectral radius of the time integration procedure for infinitely high frequencies. Numerical dissipation is active if ρ∞ < 1.0 or ξ > 0 while ρ∞ = 1.0, ξ = 0 defines the energy conserving scheme, namely the EMM by Simo & Tarnow [66]. The algorithmic energy conservation or decay can be proven analytically for the special parameter set defined by ρ∞ = 1.0, representing the EMM extended by the damping parameter ξ. For this case the analytical proof is given in Appendix A.1; for other parameter sets the conservation property is studied by means of numerical experiments in the literature (Kuhl & Crisfield [39] and Kuhl & Ramm [41]) and in section 6 of the present paper. 2.3 Spatial Discretization The spatial discretization of equation (9) is realized by subdivision of the elastic body E e B= N e=1 B and application of a standard finite element procedure to the element domain Be . Displacement-, velocity- and acceleration fields are approximated by the interpolation ¨ e . Furthermore, the variation of the matrix N and related nodal values ue , u˙ e and u Green-Lagrange strains is approximated by the deformation dependent discrete straindisplacement operator B. In order to use a general finite element notation, the components

307


of the Green-Lagrange strain tensor, the second Piola-Kirchhoff stress tensor and the constitutive tensor are contained in related matrices E, S and C of the dimension 6 × 1 or 6 × 6, respectively (for details see e.g. Zienkiewicz & Taylor [78]). u ≈ N ue ,

δu ≈ N δue ,

u˙ ≈ N u˙ e ,

¨≈Nu ë , u

δE(ue ) ≈ B(ue ) δue

(11)

These trials are introduced into the time discrete stationary condition (9) leads to the spatial and temporal discrete stationary condition of the finite element e.

δΠe (uen+1−αf ) ≈ δuen+1−αf · f edyn (uen+1 ) + f eint (uen+1 ) − f eext = 0

(12)

Since this equation holds for arbitrary variations δuen+1−αf , the following algorithmic nonlinear system of equations of the GEMM on the element level can be obtained. f edyn (uen+1 ) + f eint (uen+1 ) = f eext

(13)

In equations (12) and (13) the element vector of the effective internal forces is introduced and decomposed into algorithmic inertial forces f edyn and internal forces f eint . Furthermore, external forces f eext and the deformation independent mass matrix Me are defined. e

f dyn (uen+1 )

e

=M

f eint (uen+1 ) = f eext Me

Be

= Be

=

1−αm e 1−αm −2β e 1−αm ¨n (uen+1 − uen ) − u˙ n − u 2 β∆t β∆t 2β

BT (uen+1−αf ) (1−αf +ξ) S(uen+1 ) + (αf −ξ) S(uen ) dV NT bn+1−αf ρ dV +

(14)

NT tn+1−αf dA

∂Be

NT N ρ dV

Be

It is worth to note, that in comparison to the spatial and temporal discrete formulation of the Generalized-α Method, only the calculation of the algorithmic internal forces f eint has to be modified (see Kuhl & Crisfield [39]). 2.4 Linearization The linearization of equation (13) with respect to the displacements uen+1 yields the algorithmic incremental equation on the element level. k k k ) ∆ue = f eext − f edyn (uen+1 ) − f eint (uen+1 ) Ket (uen+1

(15)

Herein, k accounts the iteration steps within the Newton-Raphson procedure and ∆ue = k+1 k k uen+1 − uen+1 is the Newton correction of the approximated solution uen+1 . Furthermore, e the algorithmic tangent stiffness matrix Kt is defined, decomposed into the geometric stiffness matrix Keg and the material stiffness matrix Kem of the finite element. Ket (uen+1 ) = (1−αf ) Keg (uen+1 ) + (1−αf +ξ) Kem (uen+1 ) + Keg (uen+1 ) = Kem (uen+1 ) =

Be Be

1−αm Me β∆t2

BT,u (1−αf +ξ) S(uen+1 ) + (αf −ξ) S(uen ) dV BT (uen+1−αf ) C B(uen+1 ) dV

(16)

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D. Kuhl and E. Ramm

Since the discrete strain-displacement operator B is evaluated at the two configurations un+1 and un+1−αf , the material stiffness matrix Kem and consequently the tangent stiffness ˆ teaux derivative B,u requires matrix Ket are non-symmetric. The calculation of the Ga the specification of the finite element type, see for example a three dimensional continuum element by Crisfield [20] and a shell element including thickness stretch by Kuhl & Ramm [41]. Assembling the element vectors and matrices f int =

N E e=1

f eint ,

f dyn =

N E e=1

f edyn ,

f ext =

N E e=1

f eext ,

Kt =

N E e=1

Ket ,

u=

N E

ue

e=1

(17) results in the algorithmic incremental structural equation on the structural level Kt (ukn+1 ) ∆u = f ext − f int(ukn+1 ) − f dyn (ukn+1 )

(18)

which can be solved for the Newton correction ∆u. If the iterative Newton-Raphson procedure, constituted by the solution of equation (18), the update of the displacements uk+1 n+1 and a suitable convergence check, has converged, the accelerations and displacements at the end of the time step are calculated by equations (7). 3 ENHANCED ASSUMED STRAIN CONCEPT Within the framework of non-linear dynamics, the EAS concept developed by Simo & Rifai [65] extended to the geometrically non-linear case (see section 1.4) has been chosen to derive locking free finite elements. Before the algorithmic formulation of the EAS concept associated to energy controlled time integration schemes will be developed in section 4, its static formulation will be recapitulated. 3.1 Variational Formulation The key idea of the EAS concept is the additive enhancement of the compatible strain field ¯ ˜ E(u) by an independent enhanced strain field E ˜ ˜ = E(u) ¯ ˜ = 1 (F T (u) F (u) − I) + E (19) E(u, E) +E 2 introduced into the formulation of the modified Hu-Washizu functional, depending on a ˜ and an independent stress field S. ˜ displacement field u, an enhanced strain field E ˜ ˜ S) ˜ = Π(u, E,

¯ ˜ dV − W (E(u) + E)

B

˜ :E ˜ dV − S

B

¨ ) ρ dV − u · (b − u

B

u · t dA (20)

∂B

The variation of the Hu-Washizu functional, here prepared for the dynamic generalization ˜ and S ˜ ¨ ρ, with respect to the variables u, E in section 4 by the inertial force term u · u yields the stationary condition of strain enhanced elastodynamics. ˜ S) ˜ = ˜ δΠ(u, E, −

B B

¯ ¯ ˜ dV + δE(u) : S(E(u) + E) ˜ :E ˜ dV − δS

B

˜ :S ˜ dV − δE

B B

˜ : S(E(u) ¯ ˜ dV δE + E) ¨ ) ρ dV − δu · (b − u

δu · t dA = 0

∂B

(21) ¯ ˜ With the help of equation (4) the derivative of W (E(u) + E) with respect to the total strain tensor E can be understood as the second Piola-Kirchhoff stress tensor S caused ¯ ˜ by applied total strains E = E(u) + E.

309


3.2 Spatial Discretization The spatial discretization of equation (21), restricted to the static case (¨ u = 0), is performed by the finite element approximation of the displacements, its variations and the variation of the compatible strains by equation (11). The enhanced strain field is discretized by the enhanced strain interpolation matrix G and the internal enhanced strain parameters αe . ¯ e ) ≈ B(ue ) δue , E(α ˜ e ) ≈ G αe , δ E ˜ ≈ G δαe δE(u (22) ˜ is chosen to be orthogonal to that of E, ˜ see Simo & Rifai [65] for If the trial space of S the orthogonality condition, the independent stress field is eliminated in the spatial discrete form of equation (21). ˜ e (ue , αe ) ≈ δue · (f e (ue , αe ) − f eext ) + δαe · he (ue , αe ) = 0 (23) δΠ int

For arbitrary variations of displacement and enhanced strain parameters the strain extended non-linear system of equations on the element level is obtained. 

e

e

e





e



 f int (u , α )   f ext   = 

he (ue , αe )

(24)

0

The internal forces f eint , the internal forces associated to the enhanced strains he and the external forces f eext (equation (14)) in equations (23) and (24) are defined as follows. e

e

e

f int (u , α ) = he (ue , αe ) =

Be

BT (ue )S(ue , αe ) dV GT

S(ue , αe ) dV

(25)

Be

S

e

e

¯ e ) +G αe ) (u , α ) = C (E(u

3.3 Linearization The linearization of equation (24) with respect to the displacements ue and the enhanced strain parameters αe defines the extended incremental equation on the element level. 





e ek ek eT ek e  Kt (u , α ) Γ (u )   ∆u 



Γe (ue k )

He



∆αe





e e ek ek  f ext −f int (u , α ) 

=

−he (ue k , αe k )



(26)

Herein, the Newton corrections, ∆ue = ue k+1 − ue k ,

∆αe = αe k+1 − αe k

(27)

and the tangential stiffness matrix Ket , given in terms of the enhanced geometric stiffness matrix Keg and the standard material stiffness matrix Kem , are defined. Furthermore, the tangential stiffness matrices He and Γe associated to the enhanced strain parameters and their coupling to the element displacements are given: Ket (ue , αe ) = Keg (ue , αe ) + Kem (ue ) Keg (ue , αe ) = Kem (ue )

Be

= Be

BT,u S(ue , αe ) dV BT (ue ) C B(ue ) dV

Γe (ue ) = He

Be

= Be

GT C B(ue k ) dV GT C G dV

(28)

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D. Kuhl and E. Ramm

3.4 Condensation of Internal Degrees of Freedom Since the enhanced strain field is allowed to be discontinuous across the element boundaries, the incremental enhanced strain parameters ∆αe can be eliminated by static condensation. The second row of equation (26) leads to the incremental internal variables

∆αe = −He −1 Γe (ue k ) ∆ue + he (ue k , αe k )

(29)

as function of the incremental displacements. The first row of equation (26) combined with equation (29) yields the effective extended incremental equation on the element level ˜ e (ue k , αe k ) ∆ue = f e − f˜ e (ue k , αe k ) K t ext

(30)

with the tangential stiffness matrix and the internal forces: ˜ e (ue k , αe k ) = Ke (ue k , αe k ) − Γe T(ue k ) He −1 Γe (ue k ) K t t e f˜ (ue k , αe k ) = f eint (ue k , αe k ) − Γe T(ue k ) He −1 he (ue k , αe k )

(31)

˜ t , f˜ and f Assembling K ext leads to the extended effective structural equation ˜ t (uk , αk ) ∆u = f ext − f˜ (uk , αk ) K

(32)

which is solved for the incremental displacements ∆u. Finally, the enhanced strain parameters αe k+1 are updated on the element level by equations (29) and (27). 4 ALGORITHMIC FORMULATION OF THE ENHANCED ASSUMED STRAIN CONCEPT In this section the GEMM and the EAS concept are combined to the strain enhanced Generalized Energy-Momentum Method (GEMM&EAS), such that the main advantages of both methods are preserved. The modified Hu-Washizu functional and its variations (20) and (21) serve as point of departure for the development of the algorithmic formulation of the EAS concept within the framework of the energy controlled GEMM. 4.1 Temporal Discretization The temporal discretization of equation (21) is similar to the one of the pure displacement formulation given in section 2.2. In detail the following aspects have to be considered: 1. The discretization of compatible strains and their variations follow that of the total strains in the pure displacement formulation, i.e. the variation of the compatible ¯ are taken at the generalized mid-point configuration, whereas the compatstrains δE ¯ at the generalized mid-point are substituted by algorithmic compatible ible strains E strains similar to the algorithmic stress tensor in equation (9). ˜ are discretized in time similar to the compatible strains, 2. The enhanced strains E resulting in algorithmic enhanced strains. On the other hand, the variation of the ˜ is applied at the generalized mid-point configuration and is enhanced strains δE discretized in the time domain according to the displacement field in equation (6). ˜ n+1 + αf δE ñ ˜ n+1−α = (1−αf ) δE δE f

(33)

311


˜ is of no further interest, because 3. The time discretization of the incompatible stresses S the related terms will vanish during the spatial discretization due to the orthogonality condition. Therefore, only a symbolic discretization characterized by the index n+1−αf is used. 4. The time discretization of inertial, body and surface forces is not influenced by the EAS concept. In summary, the following time discrete stationary condition of the EAS method within the scope of the GEMM will be obtained. ˜ n+1−α , S ˜ n+1−α ) ˜ n+1−α , E 0 = δΠ(u f f f

= B

− −

B

+

˜ n+1−α δE f

B

δun+1−αf

˜ n+1−α : S n+1−α +ξ dV δE f f

B

δun+1−αf

+

¯ n+1−α ) : S n+1−α +ξ dV δE(u f f

1−αm 1−αm · (un+1 − un ) − u˙ n 2 β∆t β∆t ˜ n+1−α :S f

−

dV

· bn+1−αf ρ dV

−

B

B

1−αm −2β ¨ n ρ dV − u 2β

(34)

˜ n+1−α : E ˜ n+1−α dV δS f f δun+1−αf · tn+1−αf dA

∂B

Herein the algorithmic stress tensor S n+1−αf +ξ is described in terms of the compatible and enhanced strain tensor at the beginning and the end of the time step.

¯ n+1 ) + E ˜ n+1 ) + (αf −ξ) (E(u ¯ n) + E ˜ n) S n+1−αf +ξ = C : (1−αf +ξ) (E(u

(35)

Equations (34) and (35) are reduced to the standard formulation of the GEMM given by ˜ = 0, S ˜ = 0) is used. Furthermore, equation (9), if a pure displacement formulation (E the algorithmic control of total energy by the algorithmic formulation of the EAS concept is analytically proven for the special parameter set ρ∞ = 1.0 and ξ ≥ 0 in Appendix A.1. For other parameter sets numerical experiments are carried out in section 6. 4.2 Spatial Discretization In order to discretize equation (34) with respect to space, the approximations given in equations (11) and (22) as well as the orthogonality condition are introduced.

˜ e (uen+1−α , αen+1−α ) ≈ δuen+1−α · f e (uen+1 ) + f e (uen+1 , αen+1 ) − f eext δΠ int dyn f f f + δαen+1−αf

e

·h

(uen+1 , αen+1 )

(36)

=0

For arbitrary test functions δuen+1−αf and δαen+1−αf the extended non-linear equation on the element level will be obtained. 



e e e e e  f dyn (un+1 )+f int (un+1 , αn+1 ) 



he (uen+1 , αen+1 )





e  f ext 

=



0

(37)

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D. Kuhl and E. Ramm

The vector of external forces f eext is defined in equation (14) and the algorithmic inertial forces f edyn are identical to the pure displacement based formulation of the GEMM (equation (14)). The algorithmic internal forces f eint and the internal forces associated to the enhanced strains he of the EAS elements are defined as follows: f eint(uen+1 , αen+1 ) =

Be

+ e

h

(uen+1 , αen+1 ) =

Be Be

+

¯ e ) + (αf −ξ)E(u ¯ e )] dV BT (uen+1−αf ) C [(1−αf +ξ)E(u n+1 n ˜ e ) + (αf −ξ)E(α ˜ e )] dV BT (uen+1−αf ) C [(1−αf +ξ)E(α n+1 n GT

¯ e ) + (αf −ξ)E(u ¯ e )] dV C [(1−αf +ξ)E(u n+1 n

GT

˜ e ) + (αf −ξ)E(α ˜ e )] dV C [(1−αf +ξ)E(α n+1 n

(38)

Be

4.3 Linearization If equation (37) is linearized with respect to the displacements uen+1 and the enhanced strain parameters αen+1 at the end of the time step, the extended incremental equation on the element level will be obtained. 





e ek ek eT ek e  Kt (un+1 , αn+1 ) Γ (un+1−αf )  ∆u 



k ) Γe (uen+1



He

∆αe





e e ek e ek ek  f ext − f dyn (un+1 )−f int (un+1 , αn+1 ) 

=



k k , αen+1 ) −he (uen+1 (39)

In comparison to the static EAS formulation (equation (28)) and the pure displacement formulation of the GEMM (equation (16)), only the extended algorithmic geometric stiffness matrix Keg , the coupling matrix Γe and the matrix He have to be calculated in a different manner for the evaluation of the tangential in equation (39). Keg (uen+1 , αen+1 )

= Be

+

¯ e ) + (αf −ξ)E(u ¯ e )] dV BT,u C [(1−αf +ξ)E(u n+1 n ˜ e ) + (αf −ξ)E(α ˜ e )] dV BT,u C [(1−αf +ξ)E(α n+1 n

Be

Γ

e

He

(uen+1 ) n+1−αf

= (1−αf +ξ) Be

= (1−αf +ξ)

GT C B(uen+1

n+1−αf

(40) ) dV

GT C G dV

Be

4.4 Condensation of Internal Degrees of Freedom The condensation of the enhanced strain parameters in equation (39) is similar to the static formulation in section 3.4. However, the non-symmetric format of the tangential stiffness operator and the modification of the algorithmic inertial forces have to be taken into account. Consequently, the effective tangent stiffness matrix and the effective internal


313

forces are computed as follows. ˜ e (ue k , αe k ) = Ke (ue k , αe k ) K t n+1 n+1 t n+1 n+1

k k − Γe T(uen+1−α ) He −1 Γe (uen+1 ) f

e k k k k k k k k , αen+1 ) = f eint (uen+1 , αen+1 ) + f edyn (uen+1 )− Γe T(uen+1−α ) He −1 he (uen+1 , αen+1 ) f˜ (uen+1 f (41) The solution for the incremental displacements and the update of the enhanced strain parameters is based on equations (27) and (29); only the time index n + 1 has to be considered. Accelerations and velocities are updated in the classical manner by equations (7), after a converged solution of the non-linear vector equation (37) is obtained. The complete set of equations of the energy controlled algorithmic formulation utilizing the EAS method is summarized in Table 2.

5 DISCUSSION OF SIMILAR METHODS In the previous sections the combination of the EAS concept based on the enhancement of the Green-Lagrange strain tensor and the GEMM was presented. Now the question is, how this method will change, if classical time integration schemes are applied instead of the proposed time integration method or if the EAS concept is substituted by incompatible displacement models or the non-linear version of the EAS concept? In order to answer this question, the related methods are briefly discussed in the context of the combination of finite element techniques and time integration schemes. 5.1 Classical End-point Time Integration Schemes As mentioned in the introduction, classical implicit time integration schemes as the Newmark Method (Newmark [46]), the Hilber-α Method (Hilber et al. [32]), the Bossak-α Method (Wood et al. [73]) and the Generalized-α Method (Chung & Hulbert [19]) can be combined without any modification with the standard EAS method, because the calculation of the internal forces is not affected by these time marching schemes. Following the derivations of the last sections this statement can be easily proved. The combination of the EAS method and the GEMM degenerates to the strain enhanced Bossak-α Method or Newmark Method by setting αf = ξ = 0 or αf = αm = ξ = 0, respectively. The complete set of equations of the strain enhanced Newmark Method (N&EAS) is also given in Table 2. It can be observed, that the EAS terms include only static-type end-point formulations. Consequently, standard EAS finite elements are applicable within the scope of implicit time integration schemes of the Newmark and Bossak-α type. This statement can be transfered to the Generalized-α Method and the Hilber-α Method (αm = 0), if the algorithmic internal forces are calculated as a combination of the internal forces at the beginning and the end of the time step as proposed by Crisfield [21]. Since the methods developed by Hughes et al. [33] and Kuhl & Ramm [40] are based on the extension of the Newmark - or Generalized-α Method by additional constraints via Lagrange multipliers, standard EAS finite elements are applicable within the scope of the related algorithms. 5.2 Incompatible Displacement Models If the idea of incompatible displacement models by Wilson et al. [72], and Taylor et al. [71] is applied to the formulation of geometrically non-linear elastodynamics, the Green-Lagrange strain tensor can be expressed in terms of the sum of compatible and ˜. incompatible displacements u and u ˜) = E(u + u

1 T ˜ ) F (u + u ˜) − I F (u + u 2

(42)

314

D. Kuhl and E. Ramm

Generalized Energy-Momentum Method

f eint =

Newmark Method

algorithmic internal forces ¯ +E] ˜ n+1−αf +ξ dV ¯ +E] ˜ n+1 dV BT (uen+1−αf ) C [E f eint = BT (uen+1 ) C [E

Be

Be

effective algorithmic internal forces f eint + f edyn 1−αm e 1 f edyn = Me (un+1 −uen ) f edyn = Me (uen+1 −uen ) 2 2 β∆t β∆t 1−αm e 1 e 1−αm −2β e 1−2β e ¨n ¨n u˙ n + u u˙ n + u − Me − Me β∆t 2β β∆t 2β he =

Enhanced Assumed Strain algorithmic internal forces ¯ +E] ˜ n+1−αf +ξ dV ¯ +E] ˜ n+1 dV GT C [E he = GT C [E

Be

Be

algorithmic tangent stiffness 1−αm e 1 Ket = (1−αf +ξ)Kem +(1−αf )Keg + M Ket = Kem + Keg + Me 2 2 β∆t β∆t ¯ + E] ˜ n+1−αf +ξ dV ¯ + E] ˜ n+1 dV Keg = BT,u C [E Keg = BT,u C [E Be e Km =

B

T

(uen+1−αf )

C

B(uen+1 )

dV

Kem

Be

BT (uen+1 ) C B(uen+1 ) dV

=

Be

Be

Γe = (1−αf +ξ)

Be

He = (1−αf +ξ)

Enhanced Assumed Strain algorithmic tangent matrices GT CB(uen+1 ) dV Γe = GT CB(uen+1 ) dV n+1−αf

GT CG dV

He =

Be

Be

n+1

GT CG dV

Be

effective internal forces e f˜ = f eint + f edyn − Γe T (uen+1−αf ) He −1 he

e f˜ = f eint + f edyn − Γe T (uen+1 ) He −1 he

effective tangent stiffness ˜ et = Ket − Γe T (uen+1−α ) He −1 Γe (uen+1 ) K f

˜ et = Ket − Γe T (uen+1 ) He −1 Γe (uen+1 ) K

Table 2. Algorithmic and end-point formulation of Enhanced Assumed Strain finite elements

Consequently, the formulation of incompatible displacement models within the scope of the GEMM can be derived in a natural fashion from equation (9) by calculating the variation ˜ n+1−αf ) and introducing of the strain tensor at the generalized mid-point δE(un+1−αf + u ˜ n+1 ) + (αf −ξ) S(un + u ˜ n ). The spatial the algorithmic stress tensor (1−αf +ξ) S(un+1 + u discretization of the incompatible displacements is equivalent to the original formulation and the algorithmic setup is similar to that of the strain enhanced GEMM, see section 4.

315


5.3 Enhanced Assumed Strain Concept Based on the Deformation Gradient The non-linear version of the EAS concept by Simo & Armero [61] is based on the ¯ (u) and decomposition of the material deformation gradient F into the compatible part F ˜ the enhanced part F ˜) = F ¯ (u) + F ˜ = I + ∇u + F ˜ F (u, F (43) and the modified Hu-Washizu functional, depending on the displacement field u, the ˜ and the independent first Piola-Kirchhoff stress field enhanced deformation gradient F ˜ = F S. ˜ P ˜ ˜,P ˜) = Π(u, F

¯ (u)+F ˜ )) dV − W (E(F

B

˜ :F ˜ dV − P

B

¨ ) ρ dV − u · (b − u

B

u · t dA (44)

∂B

The variation of the modified Hu-Washizu functional (44) yields the stationary condition of deformation gradient enhanced elastodynamics. ˜ ˜)= δΠ(u, F˜ , P −

B

δ∇u : F S dV ˜ :F ˜ dV − δP

B

+ ˜ :P ˜ dV − δF

B

B

˜ : F S dV δF ¨ ) ρ dV − δu · (b − u

B

δu · t dA = 0

∂B

(45) ¯ (u) + F ˜ )) is Herein the constitutive second Piola-Kirchhoff stress tensor S = S(E(F defined according to equation (4). The temporal discretization of the stationary condi˜ is evaluated at the generalized mid-point tion (45) is similar to section 4.1. In fact, δΠ configuration except for the terms which determine the stresses S. ˜ n+1−α , P ˜ n+1−α ) ˜ n+1−α , F 0 = δΠ(u f f f

= − −

B

δ∇un+1−αf : F (un+1−αf ) S n+1−αf +ξ dV + δF˜ n+1−αf

B

˜ n+1−α dV :P f

δun+1−αf

−

¨ n+1−αm ρ dV − · bn+1−αf − u

B

B B

˜ n+1−α : F (un+1−α ) S n+1−α +ξ dV δF f f f ˜ n+1−α : F ˜ n+1−α dV δP f f δun+1−αf · tn+1−αf dA

∂B

(46) The algorithmic stresses S n+1−αf +ξ are defined by the displacements and the enhanced deformation gradient at the beginning and the end of the time step.

¯ (un+1 )+ F ˜ n+1 ) + (αf − ξ) E(F ¯ (un )+ F˜ n ) S n+1−αf +ξ = C : (1 − αf + ξ) E(F

(47)

˜ drop out during spatial discretization due to an equivalent Again, the extra stresses P orthogonality condition. Conservation or decay of total energy of the related time marching scheme is analytically proved for the parameter set defined by ρ∞ = 1.0 in Appendix A.1. Further evolution steps for the numerical solution of equation (46) including equation (47), as spatial discretization, linearization and condensation of the internal degrees of freedom are similar to the process described by Simo & Armero [61]. Only the modified calculation of algorithmic stresses and the evaluation of inertial terms have to be added.

316

D. Kuhl and E. Ramm

6 APPLICATION TO SHELL DYNAMICS The combination of the EAS concept and implicit energy controlled time integration schemes ¨ chter & Ramm [17] and is investigated by means of the shell element proposed by Bu ¨ chter et al. [18] including an enhanced thickness stretch. This formulation avoids Bu thickness locking and guarantees a stable time integration as well as controlled numerical dissipation of high frequencies. This shell element of course has also further modifications to avoid membrane, shear and curvature locking effects (see e.g. Bischoff & Ramm [11]). For example enhanced strain techniques (compare Table 1) may be implemented analogously while Assumed Natural Strain techniques can be applied without any algorithmic modification. However, for the sake of simplicity and perspicuity in the demonstration of the two main properties of the basic methods, only the thickness locking will be prevented within this study. The application of the shell element to selected examples of non-linear shell dynamics demonstrate the property of energy control and the improvement of the bending behavior of the EAS shell element. 6.1 Finite Shell Element Within the present study, an eight node finite shell element of the Naghdi type [44] with Reissner-Mindlin kinematics augmented by an extensible shell director field is used. It ¨ chter & Ramm [17], was originally presented for the strain enhanced static case by Bu ¨ chter et al. [18] and analyzed within the scope of the Generalized Energy-Momentum Bu Method in its pure displacement algorithmic formulation by Kuhl & Ramm [41]. The basic formulation of the shell element and the enhancement of the transverse normal strain field which avoids artificial thickness straining is briefly reviewed. For details of the shell element, the implementation of the Enhanced Assumed Strain concept and the algorithmic formulation of the one-field element see the referenced literature. The development of the shell element is based on the description of the shell space in the reference and the current configuration through the position vectors X and x connected by the displacement vector u. u = x − X = v + w θ3 ,

v = r−R ,

w = a3 − A3

(48)

Herein, v and w map the deformation from the reference to the current configuration, namely v for the shell mid-surface, characterized by the position vectors R and r, and w for the shell directors A3 and a3 . θ 3 ∈ [−1, 1] is the thickness coordinate of the shell. The shell director, the in-plane covariant base vectors of the shell mid-surface Aα , aα , α ∈ {1, 2} and the base vectors of the shell space Gi , gi , i, j ∈ {1, 2, 3} are defined by Aα = R,α ,

A3 =

A1 × A2 H , |A1 × A2 | 2

aα = r,α ,

Gi = X ,i ,

g i = x,i = ai + a3,i θ 3 (49)

where (·),i indicates the derivative with respect to the curvilinear coordinate θ i . As a ¯ = result of equations (48) and (49), the compatible Green-Lagrange strain tensor E i j 1/2 (g i · gj − Gi · Gj ) G ⊗ G is formulated in terms of the basis vectors ai and Ai . ¯ = E + +

1 [ai · aj − Ai · Aj ] Gi ⊗ Gj 2 1 [ai · a3,j + aj · a3,i − Ai · A3,j − Aj · A3,i ] θ 3 Gi ⊗ Gj 2 1 [a3,i · a3,j − A3,i · A3,j ] (θ 3 )2 Gi ⊗ Gj 2

(50)

317


ξ=0

300 200 E 100

ξ = 0.001

0

total energy

300

kinetic energy strain energy

200 E 100

N&EAS

0

40 t 60

20

80

100

0

300

300

200 E 100

200 E 100

0

0

40 t 60

20

80

100

0

GEMM&EAS

N&EAS

0

40 t 60

20

80

100

80

100

R geometry 0

40 t 60

20

ρ∞ = 1.0

ρ∞ = 0.85

Figure 2. Free motion of a rule - time histories of energy (energy E [J ] vs time t [ms])

It includes terms which are constant, linear and quadratic across the thickness, whereby quadratic terms in θ 3 may be neglected for small strain problems. The evaluation of equation (50) with i = j = 3 and A3,3 = a3,3 = 0 yields the transverse normal strain component ¯33 , which is constant across the thickness. Since for bending deformations, this strain comE ponent ought to vary linearly across the thickness due to the Poisson-coupling, the pure displacement formulation of the shell element shows an undesired stiffness effect, called Poisson- or thickness locking. In order to remedy this defect of the shell formulation, ¨ chter & Ramm [17] and Bu ¨ chter et al. [18] have proposed the extension of the Bu ˜33 within the scope strain field by a linearly varying enhanced thickness strain component E of the Enhanced Assumed Strain concept. ¯ +E ˜ =E ¯ + E˜33 G3 ⊗ G3 E=E

(51)

The discretization of the enhanced strain field is realized by a bilinear polynomial across the shell surface and a linear function in θ 3 .

˜33 = G α , E

G = θ3 1 θ1

θ2

θ1θ2 ,

αT = α1

α2

α3

α4

(52)

6.2 Free Motion of a Rule The algorithmic conservation or decay of total energy of the proposed combination of the GEMM and the EAS shell element, denoted by GEMM&EAS, will be demonstrated by the simulation of the three dimensional movement of an elastic rule. In Figure 2 the geometry, characterized through the cross section 2 mm × 60 mm and the length 300 mm, the discretization by thirty eight node, fully integrated shell elements and the applied loads are sketched. The dynamic loads increase linearly up to R(t) = 40 kN/m in the time interval t ∈ [0, 2 ms], followed by a linear decrease until zero at time t = 4 ms and vanishing

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for the remaining integration time T = 100 ms. The material data are λ = 118.8 GP a, µ = 79.2 GP a and ρ = 7800 kg/m3 . All numerical studies are performed with the constant time step ∆t = 0.05 ms. For the motion of the rule and related extensive studies on different implicit time integration schemes see Kuhl & Ramm [40, 41]. In order to point out the properties of the proposed method, the results obtained by the N&EAS and the GEMM&EAS (or the EMM&EAS) in the energy conserving and decaying versions are compared. In Figure 2 the distribution of the total, kinetic and strain energy within the integration time is plotted for different parameter sets. The EMM&EAS (GEMM&EAS for ρ∞ = 1.0 and ξ = 0) conserves the total energy in a perfect manner. In contrast to this, the time integration by the N&EAS with αm = αf = 0 and 2β = γ = 0.5 leads to the well known increase of energy and consequently to the loss of numerical stability. However, as the Newmark Method can be used with the extensively investigated standard implementation of the EAS shell element [11, 15, 16, 17, 18], the energy plateau can nevertheless be used as reference solution for the verification of the calculated energy level obtained by the proposed algorithmic concept. The controlled high frequency numerical dissipation is demonstrated by further studies with ρ∞ = 0.85 and/or ξ = 0.001. All three parameter combinations lead to a decreasing total energy. If the GEMM&EAS with ρ∞ = 0.85, ξ = 0 is compared to the related N&EAS (ρ∞ = 0.85, ξ = 0, β = 1/(ρ∞+1)2 and γ = (3−ρ∞ )/(2ρ∞ +2) [39]), it is apparent, that the energy loss obtained by the N&EAS is more distinct. This is evident, if the dissipation characteristics of both methods, described by the spectral radius as a function of time step, are considered (compare Kuhl & Ramm [41]). These results demonstrate the controllable conservation or decay of energy, and consequently the numerical stability of the proposed method. However, the improvement of the bending behavior of the finite shell element can not be illustrated by means of this example. In order to demonstrate also this property of the proposed method by the dynamic analysis of a plate in the next section is added. 6.3 Vibration of a Square Plate The objective of the present numerical experiment is the illustration of the most important numerical properties of the proposed method, namely the prevention of thickness locking, the numerical stability and the numerical dissipation of high frequencies. For this, the vibration of a square plate undergoing finite deformations is simulated. The geometry, the boundary conditions, the position and time history of the surface load, the material data and the time integration parameters are summarized in Figure 3. Due to the entered symmetry conditions only a quarter of the plate is discretized by 16 eight node fully integrated shell elements. Note, that for the three dimensional visualization of the structure the plate thickness H is magnified by the factor ten. As result of this post-processing modification, the deformation of the shell mid-surface, the thickness change and the rotation of the shell director become visible. The time interval of interest [0, T ] is subdivided by the loading phase, characterized by the ramp load, followed by the free vibration range. The bending behavior of the shell element is mainly investigated in the time interval [0, 50 ms] whereas the remaining time interval [50 ms, T ] provides information about the numerical stability, the long time and high frequency behavior of the applied time integration scheme. The detailed analysis of the loading phase is restricted to the numerical damping free EMM in the pure displacement (EMM) and strain enhanced version (EMM&EAS) and compared to the reference solution. As the Newmark Method can be combined to the standard, well investigated end-point formulation of the EAS shell element [11, 15, 16, 17, 18], the related results are used as reference solution (N&EAS), plotted with broken gray lines. The energy control in the free vibration range is studied for the EMM, the EMM&EAS and the GEMM&EAS in two parameterizations.

319


load curve

geometry, loads and boundary conditions 8

z 2m

2m

R y

R

H = 10 mm vz = 0 simply supported

vz = wx = 0 clamped

[M P a]

x

0

0

50

t [ms]

150

200

data Lamé parameters λ= 121.15 GP a µ= 80.77 GP a density ρ=7800.00 kg/m3 integration time T = 200.00 ms time step ∆t= 0.20 ms

Figure 3. Vibration of a square plate - definition of problem

In Figure 4 the motion of the plate in both characteristic time ranges, calculated by the EMM&EAS with the parameter set ρ∞ = 1.0, ξ = 0 and ∆t = 0.2 ms, is illustrated by means of sequences of deformed plates of 12 selected steps within the observed time interval. It is apparent, that a smooth dynamic response and a highly dynamic response are obtained in the loading and free vibration range, respectively. The results in the loading range calculated by the EMM, the EMM&EAS and the N&EAS are given in Figure 5. Differences of the pure displacement element and the reference solution are observable on the left side of Figure 5. The displacements obtained by the standard element are slightly smaller than the displacements calculated by the N&EAS. This is the consequence of thickness locking. In order to investigate the locking phenomenon in detail, the first six eigenvalues λ and eigenvectors φ of the linearized dynamical system (Kg + Km − λ M)φ = 0 are calculated at the end of every time step and their evolution is plotted in Figures 5 and 6. Increasing eigenvalues indicate the change of the structural behavior from a bending to membrane dominated deformation while the locking induced error of the eigenvalues is negligible for the first eigenvector, it becomes more and more significant with increasing eigenvalues. This phenomenon may be explained by the illustration of the eigenvectors in Figure 6. Since the first eigenvector represents a mode with a small curvature and consequently also small bending deformations, the differences of pure displacement and enhanced strain elements are small. Higher eigenvectors are dominated by bending deformations. Consequently, locking is activated in this deformation modes and the errors of the pure displacement model increase for higher eigenvalues. On the right hand side of Figure 5 results obtained by the EMM&EAS and the N&EAS are compared. Both, displacements and eigenvalues are identical. Therefore, the correctness of the proposed strain enhanced time stepping scheme, combining a stable, energy conserving, time integration scheme and the EAS concept avoiding thickness locking, is verified for this example. The long time behavior of the vibrating plate is illustrated by the displacement and energy plots in Figure 7. The EMM and the EMM&EAS are characterized by the conservation of total energy in the free vibration range. However, the free vibration period, observable in the displacement plots, of the EAS element is slightly larger because here the artificial stiffness pure displacement model is avoided. Furthermore, two numerical dissipative versions of the GEMM&EAS are investigated. Qualitatively, both results allow the

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t

1.6

4.8

loading phase

3.2

11.2

6.4

14.4 12.8

t

64.0

17.6

67.2

73.6

19.2

70.4 68.8

76.8 75.2

9.6

16.0

65.6 free vibration range

8.0

72.0

80.0 78.4

81.6

Figure 4. Motion of the plate in the loading and free vibration range (displacements multiplied by 3, snap-shots in time t [ms])

statement, that the total energy is decreased significantly, whereas the magnitude of the vibration, mainly characterized through the first few eigenvectors, is only weakly influenced by the numerical dissipation. This indicates, that the high frequency response is numerically dissipated, whereas the dissipation of the low frequency response is negligible since the overall response is almost not affected by the dissipation (Figure 7, left). Furthermore, it can be observed, that the period error of the energy decaying time stepping schemes is small if the solution is compared to the energy conserving version of the proposed algorithm.

321

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vertical displacements


0

0

−5 u −10

−5 u −10

−15

t

40

50

60

−15

107

107

106 λ 105

106 λ 105

104 eigenvalues(zoom)

0

EMM N&EAS 10 20

zoom 0

10

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40

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60

104

2.5

2.5

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2.0

λ 106

λ 106

0.5

0.5

0

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t

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15

20

pure displacement element (EMM)

0

snap-shots 0

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20

t

40

50

60

t

40

50

60

zoom 0

0

10

20

5

t

15

20

enhanced strain element (EMM&EAS)

Figure 5. Time histories of displacements and eigenvalues (u [cm] or λ [1/s2 ] vs t [ms])

6.4 Vibration of a Thick Square Plate In order to clarify the need of locking free finite elements in dynamic analyses, the structural response of the square plate, described in section 6.3, is again investigated, but now for thick plates. Differences of the EAS and the pure displacement shell element are emphasized by a locking sensitive thick shell geometry. The material parameters are chosen, such that the total mass and the bending stiffness are kept constant. λt = λ (H/Ht)3 Ht = µt = µ (H/Ht)3 T ρt

=

ρ H/Ht

0.10 , 0.15 m

= 60.00 ms

∆t =

(53)

0.20 ms

However, the membrane stiffness of the plate decreases with an increasing thickness. Consequently, the membrane effects emerging through the deformation of the plate and the lowest vibration frequencies are reduced. In Figure 8 the time histories of the vertical dis-

deformed t = 15.2

undeformed t = 0

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1

3 2

1

5 4

3 2

6

5 4

6

Figure 6. First six eigenvectors of the undeformed (t = 0) and the deformed (t = 15.2 ms) plate

placements, the eigenvalues and deformed shapes are shown for two different thicknesses Ht . Note, that the illustration of the shell thickness Ht is enlarged by the factor two whereas the displacements are not scaled. In contrast to the thin plate differences of the pure displacement and strain enhanced finite element are significant in the displacement and eigenvalue plots. As a results of the locking defect of the pure displacement shell element, the deformation of the plate is underestimated and the dynamical behavior is changed to a higher frequency. Consequently, the response of non-linear dynamical systems may be completely changed due to this undesired stiffness. This observation indicates, that the analysis of snap-through or mode-jumping problems requires defect free finite element discretization applicable to stable time integration schemes. In this study this objective is realized by the algorithmic formulation of the Enhanced Assumed Strain method within the framework of the Generalized Energy-Momentum Method. 7 CONCLUSION The algorithmic formulation of Enhanced Assumed Strain finite elements in the framework of energy controlled time integration schemes was presented. For this the time integration was exemplified by the Generalized Energy-Momentum Method. This special combination prevents locking in non-linear structural dynamics. The motion of the structure is numerically integrated by an energy conserving or decaying unconditionally stable time marching procedure which includes controllable numerical dissipation of high frequency structural response. The proposed method was initiated by an alternative formulation of the GEMM based on the definition of a functional, application of variational analysis and a sequential time-space discretization. Furthermore, the review of the classical EAS concept and its extension to the energy controlled time integration scheme was presented. Finally, the application of the algorithmic EAS concept was demonstrated by a shear deformable shell element including also an extensible director field. The incomplete transverse normal strain field was enriched by an enhanced strain component in order to avoid thickness locking. Examples have demonstrated, that thickness locking of the shell element is prevented

323

15 10 5 u −5 −10 −15 15 10 5 u −5 −10 −15

ρ∞ = 1.0, ξ = 0.01, EAS

ρ∞ = 1.0, ξ = 0.0, EAS

15 10 5 u −5 −10 −15

ρ∞ = 0.6, ξ = 0.0, EAS

ρ∞ = 1.0, ξ = 0.0


15 10 5 u −5 −10 −15

25 total energy

20 E 5 0

50

t

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kinetic energy 0

50

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snapshots

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0 25 20 E 5

0

50

t

150

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0 25 20 E 5

0

50

t

150

200

displacements of two selected points

0

total and kinetic energy

Figure 7. Long term behavior of the plate (u [cm] or E [kJ ] vs t [ms])

and the time integration is numerically stable. Furthermore, the total energy is conserved or decreased, depending on the user specified parameter set, whereby energy reduction is connected to high frequency dissipation. ACKNOWLEDGMENTS The present study enters research projects of collaborative research centers sponsored by the German National Science Foundation (DFG), namely project A9 at SFB 398 in Bochum and project B4 at SFB 404 in Stuttgart. Both authors gratefully acknowledge this support.

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0

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−25 u −50

−25 u −50

−75

EMM EMM&EAS t 10 20

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λ 106 2 0

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deformed shapes

0

snap-shots

10

20

t Ht = 0.1 m

40

50

10

20

t

40

50

Ht = 0.15 m

Figure 8. Time histories and deformed shapes of the thick plates (u [cm] or λ [1/s2 ] vs t [ms])

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75 Wriggers, P. and Korelc, J. (1996), “On Enhanced Strain Methods for Small and Finite Deformations of Solids”, Computational Mechanics, 18, 413-428. 76 Wriggers, P. and Reese, S. (1996), “A Note on Enhanced Strain Methods for Large Deformations”, Computer Methods in Applied Mechanics and Engineering, 135, 201-209. 77 Yeo, S.T. and Lee, B.C. (1996), “Equivalence Between Enhanced Assumed Strain Method and Assumed Stress Hybrid Method Based on the Hellinger-Reissner Principle”, International Journal for Numerical Methods in Engineering, 39, 3083-3099. 78 Zienkiewicz, O.C. and Taylor, R.L. (1989), The Finite Element Method. Volume 1 & 2, McGrawHill Book Company, London. 79 Zienkiewicz, O.C., Taylor, R.L. and Too, J.M. (1971), “Reduced Integration Technique in General Analysis of Plates and Shells”, International Journal for Numerical Methods in Engineering, 3, 275-290.

A PROOFS OF ALGORITHMIC ENERGY CONSERVATION OR DECAY The proofs of the algorithmic conservation or decay of total energy of the Generalized Energy-Momentum Method in its pure and its strain tensor or deformation gradient enhanced formulation are restricted to the special parameter set defined by ρ∞ = 1.0 and the loading free case, b = t = 0. The related time approximations given by equations (7) and (6), define the mid-point approximations of the velocity and acceleration vector 1 1 (un+1 − un ) u˙ n+ 1 = (u˙ n+1 + u˙ n ) = 2 2 ∆t (A.1) 1 2 2 1 ˙ ¨ n) = ˙ ¨ n+ 1 = (¨ ˙ un+1 + u ( u (u + u ) − = − u ) u u n+1 n n n+1 n 2 2 ∆t2 ∆t ∆t whereas the last reformulation in equation (A.1)2 is based on equation (A.1)1 . Furthermore, one preliminary calculation is necessary in order to realize the following proofs. The product δE(u) : S(u), defining the specific internal virtual work, can be substituted by ((∇δu)T F (u)) : S(u) due to the symmetry of the second Piola-Kirchhoff stress tensor. δE(u) : S(u) =

1 (∇δu)T F (u) + F T(u)(∇δu) : S(u) = (∇δu)T F (u) : S(u) (A.2) 2

A.1 Generalized Energy-Momentum Method As the time discrete formulation of the stationary condition in equation (5), including equations (A.2) and (8), has to be satisfied for admissible test functions, the mid-point velocity u˙ n+1/2 can be chosen as test function according to Simo & Tarnow [66].

(∇u˙ n+ 1 )T F (un+ 1 ) : S n+ 1 +ξ dV +

0= B

2

2

2

B

¨ n+ 1 ρ dV u˙ n+ 1 · u 2

(A.3)

2

Introducing equations (8) and (A.1) and multiplying the result by the time step ∆t yields the following expression.

0= 1 + 2

B B

1 1 (∇un+1 − ∇un )T F (un+ 1 ) : ( + ξ) S(un+1 ) + ( − ξ) S(un ) 2 2 2

(u˙ n+1 + u˙ n ) · (u˙ n+1 − u˙ n ) ρ dV

dV (A.4)

330

D. Kuhl and E. Ramm

In order to reformulate the first term of equation (A.4) some pre-calculations are necessary. The first one concerns the mid-point material deformation gradient, F (un+ 1 ) = I + ∇un+ 1 = I + 2

2

1 1 (∇un+1 + ∇un ) = (F (un+1 ) + F (un )) 2 2

(A.5)

whereby the definition of the material deformation gradient of equation (2) and the approximation of un+1/2 given in equation (6) are used. The next term can be transformed by using ∇u = F (u)−I, adding I −I and considering the definition of the Green-Lagrange strain tensor in equation (2). 1 T F (un+1 )F (un )−F T(un+1 )F (un ) 2 2 (A.6) If this term is multiplied with the symmetric second Piola-Kirchhoff stress tensor, the non-symmetric part vanishes, consequently the argument of the integral in the first term of equation (A.4) can be substituted by the scalar product of E(un+1 ) − E(un ) and the algorithmic stress tensor S n+1/2+ξ (equation (8)). Finally, further straightforward operations result in the following terms.

(∇un+1−∇un )T F (un+ 1 ) = E(un+1 )−E(un ) +

0=

1 2

B

+ξ +

1 2

B

(E(un+1 ) − E(un )) : (S(un+1 ) + S(un )) dV (E(un+1 ) − E(un )) : (S(un+1 ) − S(un )) dV

(A.7)

(u˙ n+1 + u˙ n ) · (u˙ n+1 − u˙ n ) ρ dV

B

The calculation of the first and third row and the substitution of the stresses in the second row by the constitutive law (4) yields the final form of the proof. 0=

1 2

B

+ξ +

1 2

B B

E(un+1 ) : S(un+1 ) dV −

1 2

E(un ) : S(un ) dV

B

(E(un+1 ) − E(un )) : C : (E(un+1 ) − E(un )) dV u˙ n+1 · u˙ n+1 ρ dV −

1 2

(A.8)

u˙ n · u˙ n ρ dV

B

The first two terms of equation (A.8) can be identified as the strain energies U associated with the displacements un+1 and un while the last two terms represent the kinetic energies K associated with the velocities u˙ n+1 and u˙ n . As the material tensor is positive definite, the third integral of equation (A.8) remains all times positive or zero for ξ ≥ 0. 0 = U (un+1 ) − U (un ) + K(un+1 ) − K(un )

+ξ

(E(un+1 ) − E(un )) : C : (E(un+1 ) − E(un )) dV

(A.9)

B

Consequently, the algorithmic conservation of energy is proved for ξ = 0 and energy dissipation is guaranteed for ξ > 0.

331


A.2 Enhanced Assumed Strain Extended GEMM ¯ n+1/2 : S n+1/2+ξ in equation (34) by equation (A.2), the mid-point After reformulating δE ˜ n+1/2 is substituted by the midvelocity u˙ n+1/2 is chosen as test function. Furthermore, δE ˜˙ as a suitable test function point temporal derivative of the incompatible strains E n+1/2

and discretized with respect to time by the mid-point rule in equivalence to the mid-point velocities in equation (A.1). ñ ˜ n+1 − E ˜˙ 1 = 1 E E n+ 2 ∆t

(A.10)

The enhanced stress terms can be ignored during the present proof, because they vanish during the spatial discretization due to the orthogonality condition. Therefore, the following formulation of equation (34) will be obtained after the multiplication by ∆t B

B

˜ n+1 − E ñ E

+ 1 2

1 ¯ n+1 ) + E ¯ n) + E ˜ n+1 )+( 1 − ξ)(E(u ˜ n ) dV (∇u˙ n+ 1 ) F (un+ 1 ) : C : ( + ξ)(E(u 2 2 2 2

0=

+

T

1 ¯ n+1 ) + E ¯ n) + E ˜ n+1 )+( 1 − ξ)(E(u ˜ n ) dV : C : ( + ξ)(E(u 2 2

(u˙ n+1 + u˙ n ) · (u˙ n+1 − u˙ n ) ρ dV

B

(A.11) whereby the constitutive law (4) is used in order to point out the enrichment of the compatible strains by the enhanced strains. The reformulation of the first and third term in equation (A.11) is identical to section A.1. The term associated to the enhanced strains is similarly subdivided in terms dependent and independent of ξ. 1 0= 2

B

+ξ 1 + 2

B

¯ n+1 ) − E(u ¯ n+1 ) + E ¯ n ) : C : E(u ˜ n+1 −(E(u ¯ n) + E ˜ n ) dV E(u ˜ n+1 − E ñ E

B

¯ n ) : C : E(u ˜ n+1 +(E(u ¯ n) + E ˜ n ) dV ¯ n+1 ) − E(u ¯ n+1 ) + E E(u

B

+ξ 1 + 2

ñ ˜ n+1 − E E

¯ n+1 ) + E ˜ n+1 +(E(u ¯ n) + E ˜ n ) dV : C : E(u

(A.12)

˜ n+1 −(E(u ¯ n) + E ˜ n ) dV ¯ n+1 ) + E : C : E(u

(u˙ n+1 + u˙ n ) · (u˙ n+1 − u˙ n ) ρ dV

B

If the compatible and enhanced strains are added to the total strains due to equation (19), equation (A.8) and consequently equation (A.9) will be obtained. Hence, the algorithmic conservation or decay of total energy for the presented combination of the Enhanced Assumed Strain concept and the Generalized Energy-Momentum Method is guaranteed. Alternatively, the energy conservation property of the present combination can be proved by the variation of equation (19) at the mid-point configuration. ˜ 1 ) = δE(u ¯ ˜ δE(un+ 1 , E n+ n+ 1 ) + δ E n+ 1 2

2

2

2

(A.13)

332

D. Kuhl and E. Ramm

and introduction of this result into equation (34). By introducing additionally the orthogonality condition, equation (34) is transferred to equation (9) for which the energy conservation property is studied in Appendix A.1. A.3 Enhanced Deformation Gradient Extended GEMM The proof of guaranteed energy conservation or decay of the algorithmically formulated Enhanced Assumed Strain concept by Simo & Armero [61] given by equation (46) is based on the introduction of ρ∞ = 1.0

δ∇un+ 1 + δF˜ n+ 1 : F (un+ 1 ) S n+ 1 +ξ dV +

0= B

2

2

2

2

B

¨ n+ 1 ρ dV δun+ 1 · u 2

2

(A.14)

and the variation of the mid-point material deformation gradient defined in equation (43).

˜ 1 = δ∇u 1 + δF ˜ 1 δF (un+ 1 , F˜ n+ 1 ) = δ I + ∇un+ 1 + F n+ n+ n+ 2

2

2

2

2

2

(A.15)

˜ :F ˜ and δF˜ : P ˜ cancel in equation (A.14) because of the orthogonalNote, that terms in δP ity condition. If equation (A.15) is substituted in equation (A.14) and δF : F S = δE : S is considered, equation (5) with ρ∞ = 1.0 and b = t = 0 will be obtained. As the conservation or decay of total energy for this equation is proven in Appendix A.1, this property is consequently also proved for the non-linear version of the Enhanced Assumed Strain concept.

Please address your comments or questions on this paper to: International Center for Numerical Methods in Engineering Edificio C-1, Campus Norte UPC Grand Capit´ an s/n 08034 Barcelona, Spain Phone: 34-93-4106035; Fax: 34-93-4016517 E-mail: [email protected]

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