an unsymmetric 8‐node hexahedral solid‐shell element with

Received: 27 February 2019 Revised: 19 May 2019 Accepted: 17 June 2019

DOI: 10.1002/nme.6149

R E S E A R C H A R T I C L E

An unsymmetric 8-node hexahedral solid-shell elementwith high distortion tolerance: Geometric nonlinearformulations

Zhi Li1,3 Junbin Huang1,2 Song Cen1,4 Chen-Feng Li3

1Department of Engineering Mechanics,School of Aerospace Engineering,Tsinghua University, Beijing, China2Department of Mechanical Engineering,Massachusetts Institute of Technology,Cambridge, Massachusetts3Zienkiewicz Centre for ComputationalEngineering, College of Engineering,Swansea University, Swansea, UK4AML, School of Aerospace Engineering,Tsinghua University, Beijing, China

CorrespondenceSong Cen, Department of EngineeringMechanics, School of AerospaceEngineering, Tsinghua University, Beijing100084, China; or AML, School ofAerospace Engineering, TsinghuaUniversity, Beijing 100084, China.Email: [email protected]

Funding informationNational Natural Science Foundation ofChina, Grant/Award Number: 11872229;China Scholarships Council fellowship,Grant/Award Number: 201806210280

Summary

A recent distortion-tolerant unsymmetric 8-node hexahedral solid-shell elementUS-ATFHS8, which takes the analytical solutions of linear elasticity as the trialfunctions, is successfully extended to geometric nonlinear analysis. This exten-sion is based on the corotational (CR) approach due to its simplicity and highefficiency, especially for geometric nonlinear analysis where the strain is stillsmall. Based on the assumption that the analytical trial functions can prop-erly work in each increment during the nonlinear analysis, the incrementalcorotational formulations of the nonlinear solid-shell element US-ATFHS8 arederived within the updated Lagrangian (UL) framework, in which an appropri-ate updated strategy for linear analytical trial functions is proposed. Numericalexamples show that the present nonlinear element US-ATFHS8 possesses excel-lent performance for various rigorous tests no matter whether regular or dis-torted mesh is used. Especially, it even performs well in some situations thatother conventional elements cannot work.

KEYWORDS

analytical trial function, corotational approach, finite element methods, geometric nonlinearanalysis, mesh distortion, unsymmetric solid-shell elements

1 INTRODUCTION

To date, the finite element method is still considered as the most efficient tool to simulate the complicated behaviors ofshell, one kind of the most important and complex structures in engineering, where the dimension in thickness directionis far smaller than those in the other two orthotropic directions. Generally, shell finite elements can be classified into threecategories: the classical shell elements that are based on the conventional theories of shells,1 the degenerated elements inwhich the 3D continuum is modified by some assumptions resulting in a midsurface description in analogy to standardshell theory,2-4 and the solid-shell elements formulated by directly introducing some shell features into 3D solid elementformulations.5,6 Actually, more and more researchers and users prefer to use solid-shell elements in practical simulationsbecause such models have no rotational degrees of freedom (DOFs) and can be easily applied with general 3D constitutivelaws, ie, they can seamlessly and easily connect 3D solid elements in the mesh for a complicated structure composed ofboth 3D solid and shell parts.

Int J Numer Methods Eng. 2019;1–27. wileyonlinelibrary.com/journal/nme © 2019 John Wiley & Sons, Ltd. 1

https://doi.org/10.1002/nme.6149

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2 LI ET AL.

Schoop proposed a “double-node-model” in 1986,7 which can be treated as the prototype of the kinematic descriptionmodel in solid-shell elements. In 1995, Sansour8 also developed the solid-shell concept in constructing shell elementswithout rotational DOFs. Other related works on the solid-shell elements can be found in previous works.5,6,9-11 Sincethe shell features have been considered in the formulations, the derivative solid-shell elements can be used to simulatethe behaviors of relatively thin structures. Unfortunately, conventional solid-shell model may suffer from new numericallocking problems that the degenerated shell model never encounters, such as the thickness locking. This thickness lockingis usually caused by the linear displacement interpolation in thickness direction and the coupling between in-plane andtransverse strains6 and may be handled by some techniques as summarized in the work of Huang et al.12 In addition, mostsolid-shell elements are only the simply modified versions of the conventional 3D isoparametric solid elements so thatmany defects of the original solid elements are still inevitable in the solid-shell models. Therefore, the solid-shell elementsare also susceptible to many common numerical problems13 such as the membrane locking, the transverse shear locking,the trapezoidal locking, and other difficulties caused by various mesh distortions.

Rajendran et al proposed a kind of unsymmetric finite element method immune to mesh distortion.14-17 However, someinherent defects such as interpolation failure and direction dependence exist in these formulations so that they are notconvenient and effective for practical applications.18 By introducing the analytical trial function method,19 Cen et al20

established a new unsymmetric finite element method that can overcome all original defects and improve precision. Somesuccessful models, including the plane 4-node, 8-DOF quadrilateral element US-ATFQ421 and its geometric nonlinearformulations,22 the 3D 8-node, 24-DOF hexahedral solid element US-ATFH8,23 the 8-node, 24-DOF hexahedral solid-shellelement US-ATFHS8,12 and so on,24-26 have been developed. They all exhibit much better performance than other modelsand can even perform well when other elements cannot work. The unsymmetric finite element method employs two dif-ferent sets of interpolation (test function and trial function) for displacement fields, which belongs to the Petrov-Galerkinformulation. The analytical trial function method, similar to the Trefftz methods,27 employs the solutions of govern-ing equations of elasticity as trial functions for finite element discretization. Due to the merits of these techniques, theresulting models can obtain high accuracy as well as avoid the many locking problems mentioned above. Recently, a newlocking-free 8-node unsymmetric solid-shell element US-ATFHS8 with high distortion tolerance was developed by Huanget al.12 This element was generalized from a recent 8-node, 24-DOF hexahedral solid element US-ATFH823 by introducingproper shell assumption and assumed natural strain modifications for transverse strains. It can provide highly accurateresults for linearly elastic shells with different geometries and loadings and is quite insensitive to mesh distortions. Sincemesh distortions are more common in large strain, large displacement, or large rotation problems, a distortion-tolerantfinite element model is more desirable in nonlinear analysis. However, some researchers believe that those finite elementmodels, which employ the solutions of governing equations of linear elasticity as trial functions, will be limited to theapplications of linear elastic situations.28 Actually, these analytical trial functions are the formulations composed of phys-ical coordinates with material constants. It is still possible to apply them during the nonlinear analysis if they are placedat appropriate positions and correctly updated within the current configuration at each iteration step. The topic of thispaper will focus only on the solid-shell elements for geometric nonlinear problem, one of the main nonlinear problemsfor shell structures in engineering.

During past decades, various geometric nonlinear solid-shell elements have been proposed in many works.13,29-35 Thecorotational (CR) kinematic description for nonlinear finite elements obtains more and more attention because of itssimplicity and efficiency. The configuration of an element in the CR kinematic description is usually decomposed intorigid body and deformation parts. For geometric nonlinear analysis under small strain condition, such technique allowsthose formulations from any linear elements to be directly extended to nonlinear applications. A full investigation of theCR formulations has been given in the work of Felippa and Haugen.36 Such CR description provides a chance to extendthe elements with the analytical trial functions, ie, the analytical solutions of linear elasticity, to the geometric nonlinearanalysis. However, it should also be noted that this extension is not straightforward because the feature of these analyticaltrial functions are quite different with those interpolation functions used in other elements. In this paper, the unsymmetricsolid-shell element US-ATFHS812 with analytical trial functions will be extended to geometric nonlinear analysis usinga CR kinematic description based on the updated Lagrangian framework. First, the CR frames of element US-ATFHS8are obtained by adopting the existing strategy.37 Second, the procedure of applying the updated analytical trial functionsin the CR formulations is designed, and it is the key of the whole work. Here, the incremental CR formulations must bederived within the updated Lagrangian framework so that the corresponding analytical trials functions can be updatedwithin the current configurations, while this work is not necessary for other elements using CR approach.

The content of the present work is organized as follows. In Section 2, the CR formulations based on the updatedLagrangian framework are introduced, and the best-fit CR frames for 3D continuum finite elements are briefly reviewed.

LI ET AL. 3

FIGURE 1 The motion of a body inglobal frame and local corotational frame[Colour figure can be viewed atwileyonlinelibrary.com]

In Section 3, based on a bold assumption that the analytical trial functions can properly work in each increment duringthe nonlinear analysis, the incremental CR formulations of element US-ATFHS8 for geometric nonlinear analysis areestablished, and the implementation in Abaqus via user element subroutine38 is also given. In Section 4, seven typicalgeometric nonlinear numerical examples with challenging features are used to evaluate the performance of the new non-linear element US-ATFHS8. It can be seen that the present element can provide excellent results using both regular anddistorted meshes, and it even performs well in some situations that other elements cannot work.

2 COROTATIONAL FORMULATIONS FOR GEOMETRIC NONLINEARANALYSIS

2.1 The CR formulations based on the updated Lagrangian frameworkGenerally, the motion of a body under large deformation can be decomposed into a rigid body motion part and a pure defor-mation part. This decomposition can be implemented by defining a local CR frame that continuously rotates and translateswith the element motion but does not deform with the element. Especially, in geometric nonlinear problems where thepure deformation part is small, the geometric linear theory can be used in the local CR frame. As shown in Figure 1, themotion of a general deformable body under the geometric nonlinear state is considered, and its configurations at differenttimes are expressed in both the global and the local CR frames.

Based on the updated Lagrangian framework, the equilibrium equation of the body at time t + Δt referred toconfiguration t ⌢V in the local CR frame can be derived by the principle of virtual displacements2

∫t⌢V

t+Δtt⌢S ∶ 𝛿 t+Δt

t⌢𝛆dtV −

(∫t⌢V

t+Δtt⌢f · 𝛿⌢udtV + ∫t⌢S

t+Δtt⌢t · 𝛿⌢udtS

)= 0, (1)

where t+Δtt⌢S and t+Δt

t⌢𝛆 denote the second Piola-Kirchhoff stress tensor and the Green-Lagrange strain tensor, respectively,

which are defined at time t + Δt and referred to configuration t⌢V , and

∫t⌢V

t+Δtt⌢f · 𝛿⌢udtV + ∫t⌢S

t+Δtt⌢t · 𝛿⌢udtS = 𝛿t+ΔtWext, (2)

in which 𝛿t+ΔtWext denotes the external virtual work at time t + Δt; t+Δtt⌢f and t+Δt

t⌢t denote the body force vector and

the boundary traction vector applied to configuration t+Δt ⌢V and referred to configuration t ⌢V ; 𝛿⌢u denotes the virtualdisplacement vector imposed on configuration t+Δt ⌢V .

As for the geometric nonlinear problem, we assume that the incremental displacements in the local CR frame are suffi-ciently small. Hence, Equation (1), a linear equation for solving the displacement increment vector Δ⌢u, can be rewritten

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4 LI ET AL.

FIGURE 2 The element kinematics inthe corotational formulation based onupdated Lagrangian framework [Colourfigure can be viewed atwileyonlinelibrary.com]

as

∫t⌢V𝛿 t

⌢e ∶ t⌢D ∶ t

⌢e dtV = 𝛿t+ΔtWext − ∫t⌢V

t⌢𝛔 ∶ 𝛿 t⌢e dtV (3)

by using the following relations and assumptions:

t+Δtt⌢S = t⌢𝛔 + t

⌢S ,(

tt⌢S ≡ t⌢𝛔

), (4)

{ t+Δtt⌢𝛆 = t

⌢𝛆 = t⌢e

t⌢e = 1

2

((t∇Δ⌢u

)T + t∇Δ⌢u) , (5)

t⌢S = t

⌢D ∶ t⌢e , (6)

where t⌢𝛔 is the known Cauchy stress tensor at time t in the local CR frame; t⌢S and t

⌢e are the incremental sec-ond Piola-Kirchhoff stress tensor and the linear incremental strain tensor referred to configuration t⌢V , respectively;t∇Δ⌢u = 𝜕Δ⌢u i

𝜕t⌢x 𝑗

⌢e1⌢e2 is the left gradient of the displacement increment vector Δ⌢u referred to configuration t ⌢V ; t

⌢D is theincremental stress-strain relation tensor at time t in the local CR frame.

2.2 Brief review of the best-fit CR frames for 3D continuum finite elementsTo find the CR frame is one of main procedures in the CR formulation based on the given deformable configuration. Inthis paper, the strategy to construct the best-fit CR frames for 3D continuum finite elements proposed by Mostafa andSivaselan,37 which is accomplished by minimizing deformations within the frame using a quaternion parametrizationof rotations, is adopted. Irrespective of the type of element and the number of nodes, this strategy is simple and robustwhere the minimization is stated as computing the smallest eigenvalue of a 4 × 4 matrix. The corresponding depictionand equations used in this paper are summarized in Appendices A and B.

To clarify the element kinematics in the CR formulation, a 2D element is shown in Figure 2. Based on the updatedLagrangian framework, the configuration at time t is considered as a reference configuration instead of initial configu-ration in the aforementioned work37 so that the analytical trial functions employed in present element can be updatedwithin the reference configuration. Two sets of coordinate systems are employed: the global coordinate (Cartesian coor-dinate) system and the local CR coordinate system where rotation tensor ⌢R is the relation between these two coordinatesystems. Thus, the coordinates of node I in the local CR frame t⌢x I are related to the coordinates in the global frame txIvia rotation tensor t⌢R at time t.


LI ET AL. 5

t⌢x I = t⌢RT ( txI − txO

). (7)

Then, the displacement increment vector of node I within the CR frame is

Δ⌢uI = t+Δt⌢RT ( t+ΔtxI − t+ΔtxO

)− t⌢R

T ( txI − txO), (8)

where t+ΔtxO and txO are the origin coordinates of the CR frame at time t + Δt and t, respectively. The essence of thisstrategy is to construct a CR frame ( t+Δt⌢xO,

t+Δt⌢R) by minimizing ‖Δ⌢u‖2, which can be stated as computing the smallesteigenvalue of a 4 × 4 matrix H (see Appendix A).

3 THE CONSTRUCTION OF ELEMENT US-ATFHS8 FOR GEOMETRICNONLINEAR ANALYSIS

3.1 Corotational formulations of solid-shell element US-ATFHS8In order to enhance the readability of the following finite element formulations, the notations [ ] and {} are used todenote the corresponding matrix and vector, respectively. The local CR coordinates are expressed by (⌢x , ⌢

𝑦,⌢z ). For the

components of a vector, their subscripts are expressed by (⌢x , ⌢𝑦,

⌢z ) rather than (1, 2, 3), for example, the componentsof the local displacement increments Δ⌢ui in three dimensions are (Δ⌢u⌢x , Δ⌢u⌢𝑦 , Δ⌢u⌢z ). The nodal indices are indicatedby upper case letters, for example, Δ⌢uiI is the i-component of the displacement increments at node I.

In general, the analytical trial function method that has been successfully applied in linear finite element methodcannot be carried out for geometric nonlinear finite element models because there are no closed-form analytical solutionsexisting in nonlinear problems. In the nonlinear finite element analysis, an incremental step-by-step solution is necessary.It is assumed that the solutions for the discrete time t are known while the solutions for the discrete time t + Δt need tobe determined. An approximation to the nodal displacement vector at time t + Δt is

{ t+Δtq}=

{ tq}+ {Δq} . (9)

In practice, the final nodal displacement vector at time t + Δt is usually determined by the iteration algorithm so thatEquation (9) can be rewritten as

{ t+Δtq}(k+1) = { tq} + {Δq}(k+1) =

{ t+Δtq}(k) + {dq}(k+1), (10)

where k is the iteration counter and {t+Δtq}(0) = {tq}, {Δq}(k+1) =∑k+1

i=1 {dq}(i). Similarly, the displacement, strain, andstress vectors at iteration (k + 1) can also be written as

{ t+Δtu}(k+1) =

{ t+Δtu}(k) + {du}(k+1){ t+Δt𝛆

}(k+1) ={ t+Δt𝛆

}(k) + {d𝛆}(k+1){ t+Δt𝛔}(k+1) =

{ t+Δt𝛔}(k) + {d𝛔}(k+1)

. (11)

With regard to the material nonlinear analysis, the incremental stress-strain relationship at iteration (k + 1) is

{d𝛔}(k+1) = [D](k){d𝛆}(k+1), (12)

where [D](k) is the consistent tangent modulus matrix at iteration k. Moreover, in an increment Δt, we have

{Δ𝜎}(k+1) =k+1∑i=1

[D](i−1){d𝜀}(i). (13)

6 LI ET AL.

FIGURE 3 Local corotational coordinates, 3D obliquecoordinates, and natural coordinates in a solid-shell element

For geometric nonlinear analysis where material is still linear elastic, Equation (13) can be simplified with constanttangent modulus matrix

{Δ𝛔}(k+1) = [DC]{Δ𝛆}(k+1). (14)This equation is equivalent to the stress-strain relation in linear case. Therefore, we assume that the analytical trialfunctions (solutions for linear elasticity) can properly work in each increment step

{Δu}(k+1) =[N̂](k+1) {Δq}(k+1). (15)

where [N̂](k+1) is just the shape function matrix derived from the linear analytical trial functions at iteration (k + 1). Itmeans that the analytical solutions for linear elasticity may be employed to interpolate the displacement increments ateach iteration step during the nonlinear analysis.

As shown in Figure 3, an 8-node, 24-DOF hexahedral solid-shell is considered. Nodes 1, 2,… , 8 are the corner nodes;(⌢x , ⌢

𝑦,⌢z ), (R, S, T), (𝜉, 𝜂, 𝜁) are the local CR coordinates, 3D oblique coordinates, and natural coordinates, respectively.

There are two different sets of interpolation functions for displacement fields simultaneously used with regard to theunsymmetric element method. The first set (test functions) is for the virtual displacement vector {𝛿⌢u} imposed on con-figuration t+Δt ⌢V , which is the same as that of the linear element US-ATFHS8.12 The second set (trial functions) is forthe real incremental displacement vector {Δ⌢u}. Based on the above assumption that the analytical trial functions canproperly work in each increment, the similar composite coordinates (local CR coordinates and 3D oblique coordinates)interpolation scheme is used for the displacement increment vector

in which the first twelve terms in this interpolation are related to three translational rigid motions and nine linear dis-placement fields, and the 13th∼21st terms (see Appendix C) are the displacement solutions related to the 13th∼21st stresssolutions listed in Table 1. In nonlinear formulations, the displacement solutions in 13th∼21st terms are calculated at theconfiguration t⌢V and need to be calculated at each iteration step during nonlinear analysis. Thus, the closed-form solu-tions for linear elasticity are introduced in the nonlinear analysis by doing such operation. This strategy, employing theclosed-form solutions for linear elasticity in the incremental CR formulations, is not found in other nonlinear shell ele-ments using CR approach. Furthermore, it should be noted that only the displacement increments can be approximatedby using Equation (16). Substitution of nodal coordinates and nodal displacement increments within the CR frame intoEquation (16) yields

{Δ⌢u} =⎧⎪⎨⎪⎩Δ⌢u⌢x

Δ⌢u⌢𝑦

Δ⌢u⌢z

⎫⎪⎬⎪⎭ =[ tN̂

] {Δ⌢qe}

, (17)

LI ET AL. 7

TABLE 1 Constant and linear general solutions for stress components in oblique coordinates

i 1 2 3 4 5 6 7 8 9 10 11 12Corresponding Translational Linear modesdisplacements modes

𝜎RRi

𝜎SSi

𝜎TTi 0 Constant stresses𝜎RSi

𝜎STi

𝜎RTi

i 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27Corresponding Quadratic modes (see Appendix C)displacements

𝜎RRi 0 0 0 tS 0 0 tT 0 0 0 −tR 0 0 0 −tR𝜎SSi

tR 0 0 0 0 0 0 tT 0 −tS 0 0 −tS 0 0𝜎TTi 0 tR 0 0 tS 0 0 0 0 0 0 −tT 0 −tT 0𝜎RSi 0 0 0 0 0 0 0 0 tT tR tS 0 0 0 0𝜎STi 0 0 tR 0 0 0 0 0 0 0 0 tS tT 0 0𝜎RTi 0 0 0 0 0 tS 0 0 0 0 0 0 0 tR tT

where [ tN̂]=

⎡⎢⎢⎢⎣tN̂⌢x 1

tN̂⌢x 2tN̂⌢x 3 · · · tN̂⌢x 22

tN̂⌢x 23tN̂⌢x 24

tN̂⌢𝑦 1tN̂⌢𝑦 2

tN̂⌢𝑦 3 · · · tN̂⌢𝑦 22tN̂⌢𝑦 23

tN̂⌢𝑦 24tN̂⌢z 1

tN̂⌢z 2tN̂⌢z 3 · · · tN̂⌢z 22

tN̂⌢z 23tN̂⌢z 24

⎤⎥⎥⎥⎦ , (18)

{Δ⌢qe}

=[Δ⌢u⌢x 1 Δ⌢u⌢𝑦 1 Δ⌢u⌢z 1 … Δ⌢u⌢x 8 Δ⌢u⌢𝑦 8 Δ⌢u⌢z 8

]T. (19)

Here, [ tN̂] is the shape function matrix derived from the analytical trial functions; {Δ⌢qe} is the nodal displacement

increment vector in the local CR frame. From Equation (5), the linear incremental strain tensor can be rewritten as Voigtnotation {

t⌢e}=

[ ttB̂

] {Δ⌢qe}

, (20)

where {t⌢e}=

[t⌢e ⌢x ⌢x t

⌢e ⌢𝑦 ⌢𝑦 t⌢e ⌢z ⌢z 2 t

⌢e ⌢x ⌢𝑦 2 t⌢e ⌢𝑦 ⌢z 2 t

⌢e ⌢z ⌢x]T, (21)

ttB̂ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

ttN̂⌢x 1,⌢x

ttN̂⌢x 2,⌢x

ttN̂⌢x 3,⌢x · · · t

tN̂⌢x 22,⌢xttN̂⌢x 23,⌢x

ttN̂⌢x 24,⌢x

ttN̂⌢𝑦 1,⌢𝑦


ttN̂⌢𝑦 3,⌢𝑦 · · · t

tN̂⌢𝑦 22,⌢𝑦ttN̂⌢𝑦 23,⌢𝑦


ttN̂⌢z 1,⌢z

ttN̂⌢z 2,⌢z

ttN̂⌢z 3,⌢z · · · t

tN̂⌢z 22,⌢zttN̂⌢z 23,⌢z

ttN̂⌢z 24,⌢z

ttN̂⌢x 1,⌢𝑦 + t

tN̂⌢𝑦 1,⌢xttN̂⌢x 2,⌢𝑦 + t

tN̂⌢𝑦 2,⌢xttN̂⌢x 3,⌢𝑦 + t

tN̂⌢𝑦 3,⌢x · · · ttN̂⌢x 22,⌢𝑦 + t

tN̂⌢𝑦 22,⌢xttN̂⌢x 23,⌢𝑦 + t

tN̂⌢𝑦 23,⌢xttN̂⌢x 24,⌢𝑦 + t

tN̂⌢𝑦 24,⌢x

ttN̂⌢𝑦 1,⌢z + t

tN̂⌢z 1,⌢𝑦ttN̂⌢𝑦 2,z + t

tN̂⌢z 2,⌢𝑦ttN̂⌢𝑦 3,⌢z + t

tN̂⌢z 3,⌢𝑦 · · · ttN̂⌢𝑦 22,⌢z + t

tN̂⌢z 22,⌢𝑦ttN̂⌢𝑦 23,⌢z + t

tN̂⌢z 23,⌢𝑦ttN̂⌢𝑦 24,⌢z + t

tN̂⌢z 24,⌢𝑦

ttN̂⌢z 1,⌢x + t

tN̂⌢x 1,⌢zttN̂⌢z 2,⌢x + t

tN̂⌢x 2,⌢zttN̂⌢z 3,⌢x + t

tN̂⌢x 3,⌢z · · · ttN̂⌢z 22,⌢x + t

tN̂⌢x 22,⌢zttN̂⌢z 23,⌢x + t

tN̂⌢x 23,⌢zttN̂⌢z 24,⌢x + t

tN̂⌢x 24,⌢z

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦, (22)

withttN̂iI,i =

𝜕 tN̂iI

𝜕 t⌢x i,

(i = ⌢x , ⌢𝑦, ⌢z ; I = 1, 2, · · · , 24 and ⌢x ⌢x = ⌢x , ⌢x⌢𝑦 = ⌢

𝑦,⌢x ⌢z = ⌢z

). (23)

As for the first set (test functions), no modifications are necessary because the test functions are only the function of thenatural coordinates. Thus, the virtual displacement for nonlinear solid-shell element US-ATFHS8 can be interpolatedwith the same test functions used in the linear element. Then, the virtual strain can be written as{

𝛿 t⌢e}=

[ ttB̃

] {𝛿Δ⌢qe}

, (24)

where [ ttB̃] is the strain-displacement matrix for the virtual displacement vector referred to the configuration at time t,

and it has the same expression referred to the initial configuration in reference.12 Here, we can conclude that the trialfunctions which are used to form the matrix [ t

tB̃] need be updated, while the test functions which are expressed in naturalcoordinates and used to form the matrix [ t

tB̃] need not.

8 LI ET AL.

Substitution of Equations (20) and (24) into Equation (3) yields the element stiffness matrix and the internal nodal forcevector of solid-shell element US-ATFHS8 in the local CR frame[

tt⌢K

e]= ∫t⌢V e

[ ttB̃

]T[ t⌢C]

[ ttB̂

]dV = ∫

1

−1 ∫1

−1 ∫1

−1

[ ttB̃

]T[

t⌢C] [ t

tB̂] |||[ t⌢J ]||| d𝜉d𝜂d𝜁, (25)

{tt⌢F

eint

}= ∫t⌢V e

[ ttB̃

]T { t⌢𝛔}

dtV , (26)

where [ t⌢C] and { t⌢𝛔} are the Voigt notations of the incremental stress-strain relation tensor t

⌢C and Cauchy stress tensort⌢𝛔; |[ t⌢J ]| is the Jacobian determinant in the local CR frame.

The midpoint rule is used to obtain the Cauchy stress increments, which is second-order accurate and unconditionallystable.39 Namely, the Cauchy stress increments are calculated as follows:

{Δ𝛔} ={ t+Δt∕2 ·𝛔

}Δt, (27)

where{ t+Δt∕2 ·𝛔

}is the Voigt notation of the rate of the Cauchy stress tensor at time t + Δt/2. For the isotropic elastic case,

⎧⎪⎪⎪⎨⎪⎪⎪⎩

Δ⌢𝜎⌢x ⌢x

Δ⌢𝜎⌢𝑦 ⌢𝑦

Δ⌢𝜎⌢z ⌢z

Δ⌢𝜎⌢x ⌢𝑦

Δ⌢𝜎⌢𝑦 ⌢z

Δ⌢𝜎⌢z ⌢x

⎫⎪⎪⎪⎬⎪⎪⎪⎭= E(1 − 𝜇)

(1 + 𝜇)(1 − 2𝜇)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 𝜇

1−𝜇𝜇

1−𝜇0 0 0

𝜇

1−𝜇1 𝜇

1−𝜇0 0 0

𝜇

1−𝜇𝜇

1−𝜇1 0 0 0

0 0 0 1−2𝜇2(1−𝜇)

0 0

0 0 0 0 1−2𝜇2(1−𝜇)

0

0 0 0 0 0 1−2𝜇2(1−𝜇)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎨⎪⎪⎪⎩

t⌢e ⌢x ⌢x

t⌢e ⌢𝑦 ⌢𝑦

t⌢e ⌢z ⌢z

2 t⌢e ⌢x ⌢𝑦

2 t⌢e ⌢𝑦 ⌢z

2 t⌢e ⌢z ⌢x

⎫⎪⎪⎪⎬⎪⎪⎪⎭, (28)

in which E and 𝜇 are Young's modulus and Poisson's ratio, respectively; t⌢e ij are the strain increments which are defined

as follows:

t⌢e ij =

12

(𝜕Δ⌢ui

𝜕 t+Δt∕2⌢x 𝑗

+𝜕Δ⌢u𝑗

𝜕 t+Δt∕2⌢x i

),

t+Δt∕2⌢x 𝑗 =12( t⌢x 𝑗 + t+Δt⌢x 𝑗

),

(i, 𝑗 = ⌢x , ⌢𝑦, ⌢z and ⌢x ⌢x = ⌢x , ⌢x⌢𝑦 = ⌢

𝑦,⌢x ⌢z = ⌢z

) . (29)

Since only the Cauchy stress increments can be obtained by using the above assumption and update strategy of the ana-lytical trial functions, the objective stress update algorithm should be considered to obtain the correct current Cauchystress. Owing to the linear deformation assumption in the local CR frame, the objective stress update algorithm is easy toperform here [ t+Δt⌢𝛔

]=

[ t⌢𝛔]+[Δ⌢𝛔

]. (30)

Finally, once the element stiffness matrix and the internal nodal force vector in local CR frame are obtained, theirexpressions in the global frame can be calculated using Equations (A16) and (A17) in Appendix A. The final finite elementequation can be obtained ∑

e

[ ttK

eT]{Δqe} =

∑e

{ t+ΔttR

e} , (31)

where[ t

tKeT]

is the element tangent stiffness matrix in global frame, {Δqe} is the nodal displacement increment vector inglobal frame, and { t+Δt

tRe} is the element residual vector in the global frame.

3.2 Numerical implementationThe user element subroutine of commercial software SIMULIA Abaqus38 is employed to compile and implement thepresent CR formulation of solid-shell element US-ATFHS8 for geometric nonlinear analysis. The corresponding flowchartis similar to the Figure 3 given in the work of Li et al.22 First, an input-file is written with Abaqus keywords38 to definean analysis. Then, this input-file is submitted to Abaqus/Standard, and the program is run to solve the specified problem.All results will be output by automatically writing in an Odb file. The incremental-iterative Newton-Raphson scheme

LI ET AL. 9

TABLE 2 The challenging features of seven numerical examples

Numerical example Challenging featureTwisted beam under out-of-plane load Geometric nonlinear performance of the warped

elementCantilever beam under in-plane load Membrane performance in geometric nonlinear

analysisCantilever beam subjected to end moment Bending performance in geometric nonlinear analysisSlit annular plate subjected to lifting line force Geometric nonlinear performance of element under

finite rotationsPullout of an open-ended cylindrical shell Geometric nonlinear performance of element

against bending and membrane modesPinching of a clamped cylindrical shell Geometric nonlinear performance of element

against inextensional bendingHinged cylindrical shell under concentrated load Geometric nonlinear performance of element in

capturing the post-bucking response

is employed to solve the nonlinear problem, in which the corresponding analytical trial functions are updated at eachiteration step. Moreover, all terms of the element formulation are evaluated by using a 2× 2 × 2 Gauss integration scheme.

4 NUMERICAL EXAMPLES

Seven numerical examples are presented in this section to assess the performance of the proposed formulation ofsolid-shell element US-ATFHS8 for geometric nonlinear analysis, of which the challenging features are listed in Table 2.The automatic incrementation control scheme (the size of each time increment is automatically adjusted according toconvergence) embedded in Abaqus38 is used, and the initial size of time in increment is set to 0.1 generally (the total timeis 1) due to the highly geometric nonlinearity. Results obtained by some other solid-shell and shell elements, as listedbelow, are also given for comparison.

• SC8R: An 8-node continuum shell element in Abaqus, using reduced integration with hourglass control.38

• CSS8: An 8-node continuum solid shell element in Abaqus, using incompatible modes with assumed strain, which isa new element in Abaqus since 2017.38

• S4R: A 4-node quadrilateral shell element in Abaqus, using reduced integration with hourglass control.38

• STANDER89: A 4-node quadrilateral shell element with assumed strain for finite rotation analysis proposed byStander et al.40

• PARISCH91: A 4-node quadrilateral shell element with assumed strain for finite rotation analysis proposed byParisch.41

• SZE2002: An 8-node hybrid-stress solid-shell element for geometric nonlinear analysis proposed by Sze et al.42

• SCHW2011: An 8-node solid-shell element, using reduced integration-based assumed natural strain and enhancedassumed strain approaches, for large deformation problems proposed by Schwarze and Reese.32

• MOS2013: An 8-node solid-shell element based on assumed natural deviatoric strain, assumed natural strain, andenhanced assumed strain approaches for geometric nonlinear analysis proposed by Mostafa et al.13

4.1 Twisted beam under out-of-plane loadIn this example, the geometric nonlinear performance of present element is assessed when the geometry is warped. Asshown in Figure 4, a twisted beam under out-of-plane end resultant force is considered. The geometry and materialparameters are also given in Figure 4. The linear version of this problem with thickness h= 0.32 was proposed by MacNealand Harder,43 and another version with thickness t = 0.05 proposed by Simo et al44 has a much smaller thickness thanthe former version. Therefore, the latter version is a more challenging problem, and it is used in the present example. Theonly difference is that end resultant force increases from 1 to 60. The reference results are obtained by using a fine meshof SC8R (8 × 96 elements) in Abaqus. The 4 × 24 regular mesh was used in the work of Mostafa et al.13 Here, two mesh

10 LI ET AL.

FIGURE 4 A twisted beam underout-of-plane end resultant force. A, 2 × 20regular mesh; B, 2 × 20 distorted mesh

TABLE 3 The numbers of increments NINC and iterations NITER required to obtainthe converged ultimate solutions for the twisted beam under out-of-plane end resultantforce problem (Section 4.1)

SC8R CSS8 US-ATFHS8NINC 41 35 30NITER 204 165 162

TABLE 4 The numbers of increments NINC and iterations NITER required to obtain theconverged ultimate solutions for the cantilever beam subjected in-plane end resultant forceproblem (Section 4.2)

SC8R CSS8 US-ATFHS8NINC 27 15 10NITER 109 65 49

cases, 2 × 20 regular and distorted meshes, are considered. As shown in Figure 4B, the distorted mesh is generated bychanging the z-coordinates of the nodes on the axis of the twisted beam

z̃ = z + 0.5Δz , (32)

where z is the z-coordinate in the initial regular mesh; Δz is the regular element size.The results obtained by present element US-ATFHS8, SC8R, CSS8, and the element proposed by Mostafa et al13 are

given in Figure 5. It can be seen that US-ATFHS8 can agree very well with the reference results by using both coarsemeshes (2× 20 regular and distorted meshes), which are even better than the results obtained by using a finer mesh (4× 24regular mesh) in the aforementioned work13 in x-direction and z-direction displacements. However, both SC8R and CSS8elements in Abaqus cannot provide good results using the same regular coarse mesh. It can be found that CSS8 is verysensitive to the mesh distortion; worse results appear once mesh is distorted. In addition, although SC8R is insensitive tothe mesh distortion as well as US-ATFHS8, it cannot provide good results even using regular coarse mesh.

The total numbers of increments NINC and iterations NITER required for obtaining the converged ultimate solutionsare listed in Table 3, which are obtained by using automatic incrementation control scheme as described above. In con-clusion, Figure 5 and Table 3 indicate that the performance of present element US-ATFHS8 is better than both SC8R andCSS8 because of higher accuracy and less NINC and NITER in both regular and distorted coarse meshes.

4.2 Cantilever beam under in-plane loadAs shown in Figure 6, a slender cantilever beam is subjected to in-plane end resultant force with geometric parameters,material parameters, and mesh descriptions. This example is used to evaluate the membrane performance. The thicknessdirection and load direction are set to parallel to the z-axis and y-axis, respectively. Two mesh cases, 1 × 6 elements withregular rectangle and distorted (isosceles trapezoid) in-plane shapes, are considered. The reference results are obtainedby using a fine mesh of CSS8 (4 × 8 × 240 elements) in Abaqus.

Figure 7 plots the load-displacement curves of the cantilever beam tip, which demonstrates that the present elementUS-ATFHS8 is able to present highly accurate solutions using regular and distorted meshes. On the other hand, CSS8element can give acceptable results only using regular mesh. Once the distorted mesh is used, CSS8 element will showan extremely locking phenomenon. Furthermore, SC8R element gives wrong results even when the regular mesh is usedbecause it suffers from a serious “hourglass” problem due to the reduced integration. Table 4 lists the total numbers ofincrements NINC and iterations NITER required for obtaining the converged ultimate solutions using regular mesh. Itcan be seen that the present element US-ATFHS8 has a good convergence performance as well as the accuracy.

LI ET AL. 11

FIGURE 5 Load-displacement curves for a twisted beam under out-of-plane end resultant force using 2 × 20 regular and distorted meshes(Section 4.1). A, Displacement of Point A in x-direction (UA); B, Displacement of Point A in y-direction (−VA); C, Displacement of Point A inz-direction (−WA) [Colour figure can be viewed at wileyonlinelibrary.com]

4.3 Cantilever beam subjected to end momentThis example can be used to evaluated the bending performance in the geometric nonlinear analysis,22,45,46 and it is oftenemployed for testing conventional shell and degenerated shell elements rather than solid-shell elements in literature.Here, it is employed to test the present solid-shell element US-ATFHS8. The model with geometric and material parame-ters is shown in Figure 8. The analytical solution for this problem is R = EI/M,45 where I is the section moment of inertiaand R is the radius of a circular which the cantilever beam forms. Therefore, the cantilever beam will bend to be a circlewhen the end moment M is set to 2𝜋EI/L, and the analytical solutions of the end moment against tip deflections havebeen given in the work Sze et al.45 Two mesh cases, 1 × 20 regular mesh and 2 × 20 distorted mesh, are used to assess theperformance of present element US-ATFHS8, and the results of SC8R and CSS8 in Abaqus using 2 × 40 mesh are alsogiven for comparison.

Figure 9 shows the deformed shapes obtained by US-ATFHS8, SC8R, and CSS8 using regular meshes. It should bementioned that the whole computations failed by using both SC8R and CSS8 even with 2 × 40 mesh because the solutionsappear to be diverging at around 0.21 and 0.83 step time (the whole step time is 1). However, US-ATFHS8 can present asatisfying result with both coarser meshes (1 × 20 regular elements and 2 × 20 distorted elements). The load-displacementcurves of the cantilever beam tip are also given in Figure 10.


12 LI ET AL.

FIGURE 6 A slender cantilever beamsubjected to in-plane end resultant forceand two mesh cases. A, 1 × 6 regular mesh;B, 1 × 6 distorted mesh

FIGURE 7 Load-displacement curves for a slender cantileverbeam subjected to in-plane end resultant force using 1 × 6 regularand distorted meshes (Section 4.2). A, Tip vertical displacement;B, Tip horizontal displacement [Colour figure can be viewed atwileyonlinelibrary.com]

4.4 Slit annular plate subjected to lifting line forceAs shown in Figure 11, a slit annular plate is subjected to lifting line force P at one end of the slit, and the other end ofthe slit is clamped. This example is used to assess the geometric nonlinear performance of thin-shell formulations underfinite rotations. The geometric and material parameters are also given in Figure 11. The reference results are obtained byusing 10 × 80 S4R elements, which was reported in the work of Sze et al.45 Here, the results of US-ATFHS8 using 4 × 40mesh are given to compare with the results obtained by other researchers using finer meshes13,42 in Figure 12. It shouldbe noted that the present element US-ATFHS8 can agree very well with the reference results even using a coarser mesh.


LI ET AL. 13

FIGURE 8 A slender cantilever beamsubjected to end moment and two meshcases. A, Regular mesh; B, Distorted mesh

FIGURE 9 The deformed shapes of slender cantilever beamsubjected to end moment [Colour figure can be viewed atwileyonlinelibrary.com]

FIGURE 10 Load-displacement curves for a slender cantileverbeam subjected to end moment using regular and distorted meshes(Section 4.3). A, Tip vertical displacement; B, Tip horizontaldisplacement [Colour figure can be viewed atwileyonlinelibrary.com]



14 LI ET AL.

FIGURE 11 A slit annular plate subjected to lifting line force.A, Geometry; B, Deformed shape [Colour figure can be viewed atwileyonlinelibrary.com]

FIGURE 12 Load-displacement curves for a slit annular platesubjected to lifting line force (Section 4.4) [Colour figure can beviewed at wileyonlinelibrary.com]

4.5 Pullout of an open-ended cylindrical shellIn this example, a cylindrical shell with free ends is subjected to a pair of radial force P at the middle of length. Geometricand material parameters are shown in Figure 13. Owing to symmetry, only one-eighth of the model is considered. Thereference results are also given in the work of Sze et al.45 A common mesh (8 × 12 regular elements) is used, and the16 × 24 regular and distorted meshes are also used to assess the performance of US-ATFHS8. As shown in Figure 13C,the distorted mesh is generated by changing the y-coordinates of top (bottom) nodes:

�̃� = 𝑦 + 0.5𝛼Δ𝑦, (33)

where y is the y-coordinate in the initial regular mesh; Δy is the regular element size. For different nodes with the samex and z coordinates (they are in a row), the values of 𝛼 are also same. For different nodes in two adjacent rows, the valuesof 𝛼 are taken to be 1 and − 1, respectively.



LI ET AL. 15

FIGURE 13 An open-ended cylindrical shell subjected to pulloutforce. A, Geometry; B, 8 × 12 regular mesh; C, 16 × 24 distorted mesh

FIGURE 14 Load-displacement curves for an open-endedcylindrical shell subjected to pullout force (Section 4.5) [Colourfigure can be viewed at wileyonlinelibrary.com]


16 LI ET AL.

FIGURE 15 A clamped cylindrical shell subjected to pinchingforce [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 16 Load-displacement curves for a clamped cylindricalshell subjected to pinching force (Section 4.6) [Colour figure can beviewed at wileyonlinelibrary.com]

As reported in the work of Mostafa et al,13 the response of this problem has two situations: a primary phase dominatedby bending effects and characterized by large rotations; a secondary phase dominated by membrane effects characterizedby large deformational displacements after a snap-through occurs. Figure 14 plots the results of the present elementUS-ATFHS8 in both meshes and plots the results obtained by other elements13,42 using the coarse mesh (8 × 12 elements)for comparison. It can be seen that these elements cannot agree very well with reference results when snap-through occursusing coarse mesh. The geometry discretization errors caused by the coarse mesh may be one of the reasons. ElementUS-ATFHS8 agrees very well with the reference results during the whole response using fine regular and distorted meshes(16 × 24 elements). In addition, at the maximum load, the Green strains will reach 10% or so, as reported in the work ofSze et al.42

4.6 Pinching of a clamped cylindrical shellAs shown in Figure 15, a clamped cylindrical shell is subjected to a pair of pinching force at the free end. Owing tosymmetry, only a quarter of the model is considered. This example is used to assess the trapezoidal locking problem ofsolid-shell elements when modeling curved geometries. The reference results are obtained by using a fine mesh of S4R(64 × 64 elements). A coarse mesh (16 × 16 elements) is used to evaluate the performance of US-ATFHS8. As shownin Figure 16, US-ATFHS8 can give better results than those of the solid-shell element,13 which suffers from trapezoidallocking slightly. The results of conventional shell models Parisch91 using 16 × 16 elements41 and Stander89 using 32 × 32elements40 are also given in Figure 16. It can be seen that US-ATFHS8 does not have any trapezoidal locking problemeven using a coarser mesh.

4.7 Hinged cylindrical shell under concentrated loadThis is a common postbucking geometric nonlinear example. As shown in Figure 17, a segment of a cylindrical shell ishinged at the two edges and subjected to a concentrated load. Owing to symmetry, only a quarter of the model is consid-ered. There are two versions of this problem which has been used in the literature. The work of Legay and Combescure47



LI ET AL. 17

FIGURE 17 A hinged cylindrical shellunder concentrated load. A, Geometry;B, Regular mesh; C, Distorted mesh

FIGURE 18 Load-displacement curves for a hinged cylindricalshell under concentrated load (Section 4.7) [Colour figure can beviewed at wileyonlinelibrary.com]

considered a thick shell with thickness h = 12.7, while Cardoso et al30,48 used a thin shell with thickness h = 6.35. In thepresent work, the latter one is chosen because it is a more challenging case. The reference results are obtained with a finemesh of SC8R (64 × 64 × 2 elements). As reported in the work of Schwarze and Reese,32 the thickness is discretized withtwo elements when using solid-shell element to represent the hinged support. Two coarse meshes (4 × 4 × 2 regular and


18 LI ET AL.

distorted elements) are used in the present work. In order to study the postbucking nonlinear behavior of the frame atpoints A and B, the modified Riks method4,38 is employed here.

Similar to previous examples, the automatic incrementation control scheme is chosen, and the initial increment size inarc length along the static equilibrium pathΔlin, the maximum arc length incrementΔlmax, and the maximum value of theload proportionality factor 𝜆end are set to 0.05, 0.1, and 1, respectively. The load-displacement curves of US-ATFHS8 andother models13,32 are plotted in Figure 18 for comparison. In general, US-ATFHS8 can agree well with the reference resultsusing both regular and distorted coarse meshes. The deviation appears when snap-through occurs at point A mainlybecause the mesh is quite coarse such that the geometry discretization errors apply. On the other hand, no deviationappears when snap-through occurs at point B. Therefore, it can be said that US-ATFHS8 can capture the postbuckingbehavior in geometric nonlinear analysis using both regular and distorted coarse meshes.

5 CONCLUSIONS REMARKS

Based on the assumption that the analytical trial functions can properly work in each increment during the nonlinearanalysis and a simple and efficient CR approach, a recent distortion-tolerant unsymmetric 8-node hexahedral solid-shellelement US-ATFHS812 is successfully extended to geometric nonlinear analysis. First, the incremental CR approachis adopted, and the corresponding CR frame are obtained by using the strategy proposed in the work of Mostafa andSivaselvan.37 Then, the assumption that the closed solutions for linear elasticity can be used in the nonlinear analy-sis is proposed. Subsequently, the incremental CR formulations of element US-ATFHS8 are derived within the updatedLagrangian framework so that the analytical trial functions employed in this element can be updated during geometricnonlinear analysis. However, this work is not necessary for other elements using CR approach. Since only the Cauchystress increments can be obtained by using the above assumption and update strategy of the analytical trial functions,the objective stress update algorithm should be considered to obtain the correct current Cauchy stress, and it is easy toperform using CR approach. Compared with other shell elements based on the CR approach, the main feature of thiselement is the employment of the updated trial functions.

Numerical examples show that the present nonlinear element exhibits excellent performance for challenging geomet-ric nonlinear problems no matter whether regular or distorted mesh is used. Especially, it even performs well in somesituations that other elements cannot work, as described in Section 4.3. It not only verifies the proposed assumption thatthe analytical trial functions can properly work in each increment during the nonlinear analysis but also demonstratesthe advantages of the unsymmetric finite element method with analytical trial functions, although these functions onlycome from linear elasticity.

How to generalize the unsymmetric element with linear analytical solutions to material nonlinear analysis is anotherinteresting topic. The key point is whether the corresponding analytical trial functions can properly work in each incre-ment under material nonlinear situations including the finite strain cases. Related issues will be discussed in the nearfuture.

ACKNOWLEDGEMENTS

The authors would like to thank for the financial supports from the National Natural Science Foundation of China undergrant 11872229 and the China Scholarships Council fellowship under grant 201806210280 during the visit of Zhi Li toSwansea University.

ORCID

Zhi Li https://orcid.org/0000-0003-2495-5816Junbin Huang https://orcid.org/0000-0002-6839-286XSong Cen https://orcid.org/0000-0002-8674-4005

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https://orcid.org/0000-0003-2495-5816

https://orcid.org/0000-0003-2495-5816

https://orcid.org/0000-0002-6839-286X

https://orcid.org/0000-0002-6839-286X

https://orcid.org/0000-0002-8674-4005

https://orcid.org/0000-0002-8674-4005

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Eng. 2003;57(9):1299-1322.48. Cardoso RP, Yoon JW, Mahardika M, Choudhry S, Alves de Sousa R, Fontes Valente R. Enhanced assumed strain (EAS) and assumed

natural strain (ANS) methods for one-point quadrature solid-shell elements. Int J Numer Methods Eng. 2008;75(2):156-187.49. Weng J, Huang TS, Ahuja N. Motion and structure from two perspective views: algorithms, error analysis, and error estimation. IEEE

Trans Pattern Anal Mach Intell. 1989;11(5):451-476.50. Horn BK. Closed-form solution of absolute orientation using unit quaternions. JOSA A. 1987;4(4):629-642.51. Yuan KY, Huang YS, Yang HT, Pian THH. The inverse mapping and distortion measures for 8-node hexahedral isoparametric elements.

Computational Mechanics. 1994;14(2):189-199.

How to cite this article: Li Z, Huang J, Cen S, Li C-F. An unsymmetric 8-node hexahedral solid-shell ele-ment with high distortion tolerance: Geometric nonlinear formulations. Int J Numer Methods Eng. 2019;1–27.https://doi.org/10.1002/nme.6149

APPENDIX A

THE BEST-FIT COROTATIONAL FRAME37

A.1 Best-fit corotational frameBased on Equation (8), the displacement increment vector of the element with n nodes in the CR frame can be written as

Δ⌢u = diag(

t+Δt⌢R)T ( t+Δtx − stack

( t+ΔtxO))

− diag( t⌢R)T ( tx − stack( txO

)), (A1)

where 𝛥⌢u = [𝛥⌢uT

1 , · · · , 𝛥⌢uTn]T, t+Δtx = [ t+ΔtxT

1 , · · · ,t+ΔtxT

n]T, tx = [ txT1 , · · · ,

txTn]T, stack(t+ΔtxO), and stack(txO) are vec-

tors formed by stacking up n copies of t+ΔtxO and txO. The strategy is to construct the CR frame ( t+Δt⌢xO,t+Δt⌢R) so as to

minimize ‖𝛥⌢u‖2, which is referred as a best-fit CR frame.

https://doi.org/10.1002/nme.6149

LI ET AL. 21

Let t+ΔtxC and txC be the element centroids in the deformed and reference configurations. We have

‖Δ⌢u‖2 =n∑

I=1

‖‖‖Δ⌢uI‖‖‖2

=n∑

I=1

‖‖‖‖ t+Δt⌢RT ( t+ΔtxI − t+ΔtxO

)− t⌢R

T ( txI − txO)‖‖‖‖2

=n∑

I=1

‖‖‖‖( t+ΔtxI − t+ΔtxO)− t+Δt⌢R t⌢R

T ( txI − txO)‖‖‖‖2

=n∑

I=1

‖‖‖‖( t+ΔtxI − t+ΔtxC)−( t+ΔtxO − t+ΔtxC

)− t+Δt⌢R t⌢R

T (( txI − txC)−( txO − txC

))‖‖‖‖=

n∑I=1

‖‖‖dI − t+Δt⌢RcI − xO‖‖‖2

=n∑

I=1

‖‖‖dI − t+Δt⌢RcI‖‖‖2

+ n‖‖xO‖‖2

, (A2)

wheredI = t+ΔtxI − t+ΔtxC

cI = t⌢RT ( txI − txC

)xO = t+ΔtxO − t+ΔtxC + t+Δt⌢R t⌢R

T ( txO − txC). (A3)

To minimize ‖Δ⌢u‖2, n‖xO‖2 should be set to 0. Therefore, we have t+ΔtxO = t+ΔtxC by assuming that txO has been chosento be the centroid in the reference configuration, ie, txO = txC. Then, finding the best-fit CR frame reduces to

t+Δt∼⌢R = argmin

t+Δt⌢R

n∑I=1

‖‖‖dI − t+Δt⌢RcI‖‖‖2

or argmint+Δt⌢R

‖‖‖‖D − diag(

t+Δt⌢R)

C‖‖‖‖2, (A4)

where D = [dT1 , · · · ,d

Tn]T, and C = [cT

1 , · · · , cTn]T.

A quaternion parametrization of rotations is used to transform (A4) into an eigenvalue problem which follows the workof Weng et al.49 The relevant concepts and notation related to quaternions are given in Appendix B. We start with thesummand in Equation (A4) ‖‖‖dI − t+Δt⌢RcI

‖‖‖2=

‖‖‖‖ ◦dI −Q

◦cIQ

∗‖‖‖‖2

=‖‖‖‖ ◦

dIQ −Q◦cI‖‖‖‖

2

=‖‖‖‖‖([

◦dI

]L−[◦cI

]R

)Q‖‖‖‖‖

2

= ‖𝛽Q‖2

, (A5)

where

𝛽 =[◦dI

]L−[◦cI

]R, (A6)

Thus, (A4) can be restated asmin QTHQ

subject to QTQ = 1, (A7)

where

H =n∑

I=1𝛽T𝛽. (A8)

22 LI ET AL.

Therefore, the minimum value is the smallest eigenvalue of the 4 × 4 matrix H, which is denoted as ∼𝜆, and the minimizer

is the corresponding normalized eigenvector Q̃. Since H is symmetric positive semidefinite and has a full eigenspace and�̃�, the minimum value of ‖Δ⌢u‖2 is nonnegative. We thus have

t+Δt∼⌢R = Rq

( ∼Q). (A9)

A.2 Linearization of kinematicsEquation (A1) can be written as

Δ⌢u = diag(

t+Δt∼⌢R)T

D − C . (A10)

Linearizing this equation gives

DΔ⌢u · 𝛿D = diag(

D t+Δt∼⌢R · 𝛿D

)T

D + diag(

t+Δt∼⌢R)T

𝛿D. (A11)

Because (D t+Δt∼⌢R ·𝛿D) t+Δt

∼⌢R

Tis linear in 𝛿D, we conclude that there is a matrix G such that (D t+Δt

∼⌢R ·𝛿D) t+Δt

∼⌢R

T= (G𝛿D)=

(see Appendix B for the notation ()=). The matrix G can be interpreted as extracting the instantaneous rotation from aninstantaneous motion 𝛿D. Thus,

D t+Δt∼⌢R · 𝛿D = (G𝛿D)= t+Δt

∼⌢R, (A12)

and

diag(

D t+Δt∼⌢R · 𝛿D

)T

D = −diag(

t+Δt∼⌢R

T(G𝛿D)=

)D

= −

⎧⎪⎪⎨⎪⎪⎩t+Δt

∼⌢R

T(G𝛿D)=d1⋮

t+Δt∼⌢R

T(G𝛿D)=dn

⎫⎪⎪⎬⎪⎪⎭= −

⎧⎪⎪⎨⎪⎪⎩

t+Δt∼⌢R

T (−d1

)G𝛿D

⋮

t+Δt∼⌢R

T (−dn

)G𝛿D

⎫⎪⎪⎬⎪⎪⎭= −diag

(t+Δt

∼⌢R)T

SG𝛿D

, (A13)

where

S =

⎧⎪⎪⎨⎪⎪⎩−d

T

1

⋮

−dT

n

⎫⎪⎪⎬⎪⎪⎭. (A14)

The three columns of S are the instantaneous motions of the nodes in the deformed configuration, when unit instanta-neous rotations are applied about axes at the origin of the CR frame parallel to the global x, y, and z axes, respectively.Thus, we have

DΔ⌢u · 𝛿D = diag(

t+Δt∼⌢R)T

(I − SG)𝛿D. (A15)

Finally, the global nodal force vector and stiffness matrix can be expressed in terms of the forms in the local CR frame:

fglobal = (I − SG)Tdiag(

t+Δt∼⌢R)

flocal, (A16)

LI ET AL. 23

Kglobal = Kmat + Kgeo

Kmat = (I − SG)Tdiag(

t+Δt∼⌢R)

Klocaldiag(

t+Δt∼⌢R)T

(I − SG)

Kgeo = GTSfT(

I − 12

SG)+(

I − 12

SG)T

Sf G

, (A17)

where

Sf =⎧⎪⎨⎪⎩−fglobal 1

T

⋮

−fglobal n

T

⎫⎪⎬⎪⎭ . (A18)

APPENDIX B

QUATERNIONS

The concepts of quaternions used in this paper37 are summarized as follows. A detailed description of quaternions andthe relations to rotations can be found in the works of Weng et al49 and Hong.50

Firstly, the notation related to the vector cross product is introduced. If a = (a1, a2, a3)T is a vector, a (or a=) denotes theskew-symmetric matrix ⎡⎢⎢⎣

0 −a3 a2

a3 0 −a1

−a2 a1 0

⎤⎥⎥⎦ , (B1)

representing cross product by a. Thus, for a vector b, we have ab = a × b.Quaternions are a generalization of complex numbers and are a convenient way to represent rotations. A quaternion Q

consists of a scalar component q0 and a vector component q and is written as Q = (q0, q). The conjugate of a quaternionQ is defined as Q* = (q0,−q). Multiplication of quaternions is defined by

PQ = (p0q0 − p · q, p0q + q0p + p × q). (B2)

The operation is not commutative but is associative.If quaternions are thought of as 4-vectors, multiplication from the left by quaternion P has the matrix representation

[Q]L =

[q0 −qT

q q0I + q

](B3)

so that QP = [Q]LP. Similarly, multiplication from the right has the matrix representation

[Q]R =

[q0 −qT

q q0I − q

], (B4)

that is, PQ = [Q]RP. [Q]L and [Q]R are both orthogonal matrices. We also note that the matrix representations ofmultiplication by quaternion conjugates are the transposes,

Q∗P = [Q]LTP

PQ∗ = [Q]RTP

. (B5)

For a vector a, the quaternion (0, a) is called a pure imaginary quaternion and is denoted by◦a. Q is called a unit

quaternion if QQ∗ = Q∗Q = q20 + ‖q‖2 = 1. The operation Q

◦aQ∗ of a unit quaternion on a vector a is called conjugation,

24 LI ET AL.

which corresponds to a rotation about an axis in the direction of q by an angle 2cos−1q0. We also note that the two unitquaternions ±Q represent the same rotation. The matrix representation of conjugation is

Q◦aQ∗ =

([Q]L

◦a)Q∗ = [Q]R

T([Q]L

◦a)=

([Q]R

T[Q]L) ◦

a, (B6)

and then, we note that

[Q]RT[Q]L =

⎡⎢⎢⎢⎢⎣1 0 0 00 q2

0 + q21 − q2

2 − q23 2 (q2q1 − q0q3) 2 (q3q1 + q0q2)

0 2 (q2q1 + q0q3) q20 − q2

1 + q22 − q2

3 2 (q3q2 − q0q1)0 2 (q3q1 − q0q2) 2 (q3q2 + q0q1) q2

0 − q21 − q2

2 + q23

⎤⎥⎥⎥⎥⎦. (B7)

Therefore, a unit quaternion Q is associated with the rotation matrix

Rq(Q) =⎡⎢⎢⎢⎣

q20 + q2

1 − q22 − q2

3 2 (q2q1 − q0q3) 2 (q3q1 + q0q2)2 (q2q1 + q0q3) q2

0 − q21 + q2

2 − q23 2 (q3q2 − q0q1)

2 (q3q1 − q0q2) 2 (q3q2 + q0q1) q20 − q2

1 − q22 + q2

3

⎤⎥⎥⎥⎦ , (B8)

where (q1, q2, q3) are the components of the vector q.

APPENDIX C

ANALYTICAL GENERAL SOLUTIONS FOR QUADRATIC DISPLACEMENTS IN TERMS OFOBLIQUE COORDINATES

The relationship between the global Cartesian coordinates and natural coordinates is given by

{ x𝑦

z

}=

8∑i=1

Ni

{ xi𝑦izi

}=

⎧⎪⎨⎪⎩x0 + a1𝜉 + a2𝜂 + a3𝜁 + a4𝜉𝜂 + a5𝜂𝜁 + a6𝜉𝜁 + a7𝜉𝜂𝜁

𝑦0 + b1𝜉 + b2𝜂 + b3𝜁 + b4𝜉𝜂 + b5𝜂𝜁 + b6𝜉𝜁 + b7𝜉𝜂𝜁

z0 + c1𝜉 + c2𝜂 + c3𝜁 + c4𝜉𝜂 + c5𝜂𝜁 + c6𝜉𝜁 + c7𝜉𝜂𝜁

⎫⎪⎬⎪⎭ , (C1)

in which

Ni =18(1 + 𝜉i𝜉) (1 + 𝜂i𝜂) (1 + 𝜁i𝜁 ) , i = 1, 2, … , 8 (C2)

are the shape functions of the 8-node tri-linear isoparametric element, and

⎧⎪⎨⎪⎩x0

𝑦0

z0

⎫⎪⎬⎪⎭ = 18

8∑i=1

⎧⎪⎨⎪⎩xi

𝑦i

zi

⎫⎪⎬⎪⎭ ,

⎧⎪⎨⎪⎩a1

b1

c1

⎫⎪⎬⎪⎭ = 18

8∑i=1

𝜉i

⎧⎪⎨⎪⎩xi

𝑦i

zi

⎫⎪⎬⎪⎭⎧⎪⎨⎪⎩a2

b2

c2

⎫⎪⎬⎪⎭ = 18

8∑i=1

𝜂i

⎧⎪⎨⎪⎩xi

𝑦i

zi

⎫⎪⎬⎪⎭ ,

⎧⎪⎨⎪⎩a3

b3

c3

⎫⎪⎬⎪⎭ = 18

8∑i=1

𝜁i

⎧⎪⎨⎪⎩xi

𝑦i

zi

⎫⎪⎬⎪⎭, (C3)

where 𝜉i, 𝜂i, 𝜁 i, xi, yi, and zi are the natural coordinates and global coordinates at each node, respectively.

LI ET AL. 25

The linear relationship between oblique coordinates and Cartesian coordinates is determined by the Jacobian matrixJ0 at the origin of the natural coordinate system51

⎧⎪⎨⎪⎩R

S

T

⎫⎪⎬⎪⎭ =(J−1

0)T

⎛⎜⎜⎜⎝⎧⎪⎨⎪⎩

x

𝑦

z

⎫⎪⎬⎪⎭ −⎧⎪⎨⎪⎩

x

𝑦

z

⎫⎪⎬⎪⎭𝜉=𝜂=𝜁=0

⎞⎟⎟⎟⎠ =1J0

⎡⎢⎢⎣a1 b1 c1

a2 b2 c2

a3 b3 c3

⎤⎥⎥⎦⎧⎪⎨⎪⎩

x − x0

𝑦 − 𝑦0

z − z0

⎫⎪⎬⎪⎭ , (C4)

J0 =

⎡⎢⎢⎢⎢⎢⎣

𝜕x𝜕𝜉

𝜕𝑦

𝜕𝜉

𝜕z𝜕𝜉

𝜕x𝜕𝜂

𝜕𝑦

𝜕𝜂

𝜕z𝜕𝜂

𝜕x𝜕𝜁

𝜕𝑦

𝜕𝜁

𝜕z𝜕𝜁

⎤⎥⎥⎥⎥⎥⎦𝜉=𝜂=𝜁=0

=

⎡⎢⎢⎢⎢⎣a1 b1 c1

a2 b2 c2

a3 b3 c3

⎤⎥⎥⎥⎥⎦, (C5)

J0 = |J0| = a1

(b2c3 − b3c2

)+ a2

(b3c1 − b1c3

)+ a3

(b1c2 − b2c1

)= a1a1 + a2a2 + a3a3 = b1b1 + b2b2 + b3b3 = c1c1 + c2c2 + c3c3

a1 = b2c3 − b3c2, b1 = a3c2 − a2c3, c1 = a2b3 − a3b2



. (C6)

Denote

h1 = b2c3 + b3c2, h2 = a2c3 + a3c2, h3 = a2b3 + a3b2



. (C7)

The work of Zhou et al23 has provided related solutions for both isotropic and anisotropic materials. Here, onlythe 13th∼21st isotropic displacement solutions used in Equation (16) are listed as follows, in which Axi, Ayi, Azi, Axyi,Ayzi, Azxi, (i = 13,… , 21) are the corresponding parameters associated with the material and geometry defined by theaforementioned work.23.

1. The 13th∼15th sets of solutions for displacements (i = 13∼15)

Ui =1

2J0

{[a1J0Axi +

(J0 − a1a1

) (a1Axi + b1Axyi + c1Azxi

)− a1

(b

21Ayi + c2

1Azi + b1c1Ayzi

)]R2

− a1

(a2

2Axi + b22Ayi + c2

2A𝑧𝑖 + a2b2Axyi + b2c2A𝑦𝑧𝑖 + a2c2Azxi

)S2

− a1

(a2

3Axi + b23Ayi + c2

3Azi + a3b3Axyi + b3c3Ayzi + a3c3Azxi

)T2

+[

J0

(2a2Axi + b2Axyi + c2Azxi

)− 2a1

(a1a2Axi + b1b2Ayi + c1c2Azi

)− a1

(h9Axyi + h7Ayzi + h8Azxi

)]RS

+[

J0


)− 2a1


)− a1

(h6Axyi + h4Ayzi + h5A𝑧𝑥𝑖

)]RT

− a1

(2a2a3Axi + 2b2b3Ayi + 2c2c3Azi + h3Axyi + h1Ayzi + h2Azxi

)ST

}(C8a)

Vi =1

2J0

{[b1J0Ayi +

(J0 − b1b1

)(a1Axyi + b1Ayi + c1Ayzi

)− b1

(a2

1Axi + c21Azi + a1c1Azxi

)]R2

− b1

(a2

2A𝑥𝑖 + b22Ayi + c2


)S2

− b1

(a2

3Axi + b23Ayi + c2


)T2

+[

J0

(a2Axyi + 2b2Ayi + c2A𝑦𝑧𝑖

)− 2b1


)− b1


)]RS

+[

J0

(a3Axyi + 2b3Ayi + c3Ayzi

)− 2b1


)− b1


)]RT

− b1


)ST

}(C8b)

26 LI ET AL.

Wi =1

2J0

{[c1J0Azi +

(J0 − c1c1

) (a1Azxi + b1Ayzi + c1Azi

)− c1

(a2

1Axi + b21Ayi + a1b1Axyi

)]R2

− c1

(a2

2Axi + b22Ayi + c2


)S2

− c1

(a2

3Axi + b23Ayi + c2


)T2

+[

J0

(a2Azxi + b2Ayzi + 2c2Azi

)− 2c1


)− c1


)]RS

+[

J0


)− 2c1


)− c1


)]RT

− c1


)ST

}. (C8c)

2. The 16th∼18th sets of solutions for displacements (i = 16∼18)

Ui =1

2J0

{−a2

(a2

1Axi + b21Ayi + c2

1 Azi + a1b1Axyi + b1c1Ayzi + a1c1Azxi

)R2

+[

a2J0Axi +(

J0 − a2a2) (

a2Axi + b2Axyi + c2Azxi

)− a2

(b

22Ayi + c2

2Azi + b2c2Ayzi

)]S2

− a2

(a2

3Axi + b23 Ayi + c2


)T2

+[

J0


)− 2a2

(a1a2Axi + b1b2Ayi + c1c2 Azi

)− a2


)]RS

− a2

(2a1a3Axi + 2b1b3 Ayi + 2c1c3 Azi + h6Axyi + h4Ayzi + h5Azxi

)RT

+[

J0


)− 2a2


)− a2


)]ST

}

(C9a)

Vi =1

2J0

{−b2

(a2

1Axi + b21Ayi + c2


)R2

+[

b2J0Ayi +(

J0 − b2b2

)(a2Axyi +b2Ayi + c2Ayzi

)− b2

(a2

2Axi + c22Azi + a2c2Azxi

)]S2

− b2

(a2

3Axi + b23Ayi + c2


)T2

+[

J0


)− 2b2


)− b2


)]RS

− b2


)RT

+[

J0


)− 2b2


)− b2


)]ST

}

(C9b)

Wi =1

2J0

{−c2

(a2

1Axi + b21Ayi + c2


)R2 +

[c2J0Azi +

(J0 − c2c2

) (a2Azxi

+ b2Ayzi + c2Azi

)− c2

(a2

2Axi + b22Ayi + a2b2Axyi

)]S2 − c2

(a2

3Axi + b23Ayi + c2

3 Azi + a3b3Axyi + b3c3Ayzi

+ a3c3Azxi

)T2 +

[J0


)− 2c2


)− c2

(h9Axyi + h7Ayzi

+ h8Azxi

)]RS − c2


)RT +

[J0

(a3Azxi + b3Ayzi

+ 2c3Azi

)− 2c2


)− c2


)]ST

}. (C9c)

LI ET AL. 27

3. The 19th∼21st sets of solutions for displacements (i = 19∼21)

Ui =1

2J0

{−a3

(a2

1Axi + b21 Ayi + c2


)R2 − a3(a

22Axi + b

22 Ayi + c2

2 Azi

+ a2b2Axyi + b2c2Ayzi + a2c2Azxi )S2 +[

a3J0Axi +(

J0 − a3a3) (

a3Axi + b3Axyi + c3Azxi

)− a3

(b

23Ayi

+ c23Azi + b3c3Ayzi

)]T2 − a3


)RS

+[

J0


)− 2a3


)− a3


)]RT

+[

J0


)− 2a3


)− a3


)]ST

}(C10a)

Vi =1

2J0

{−b3

(a2

1Axi + b21Ayi + c2


)R2 − b3

(a2

2Axi + b22 Ayi + c2

2 Azi

+ a2b2Axyi + b2c2Ayzi + a2c2Azxi

)S2 +

[b3J0Ayi +

(J0 − b3b3

)(a3Axyi + b3Ayi + c3Ayzi

)− b3

(a2

3Axi

+ c23Azi + a3c3Azxi

)]T2 − b3


)RS

+[

J0


)− 2b3


)− b3


)]RT

+[

J0


)− 2b3


)− b3


)]ST

}(C10b)

Wi =1

2J0

{−c3

(a2

1Axi + b21 Ayi + c2


)R2 − c3

(a2

2Axi + b22 Ayi + c2

2 Azi

+ a2b2Axyi + b2c2Ayzi + a2c2Azxi

)S2 +

[c3J0Azi +

(J0 − c3c3

) (a3Azxi + b3Ayzi + c3Azi

)− c3

(a2

3Axi

+ b23Ayi + a3b3Axyi

)]T2 − c3


)RS

+[

J0


)− 2c3


)− c3


)]RT

+[

J0


)− 2c3


)− c3


)]ST

}. (C10c)

an unsymmetric 8‐node hexahedral solid‐shell element with

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