INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng (2014) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4765
NURBS based least-squares finite element methods for fluid and solid mechanics C. Kadapa, W.G. Dettmer and D. Peri´c*,† College of Engineering, Civil and Computational Research Centre, Swansea University, Singleton Park, Swansea SA2 8PP, UK
SUMMARY This contribution investigates the performance of a least-squares finite element method based on nonuniform rational B-splines (NURBS) basis functions. The least-squares functional is formulated directly in terms of the strong form of the governing equations and boundary conditions. Thus, the introduction of auxiliary variables is avoided, but the order of the basis functions must be higher or equal to the order of the highest spatial derivatives. The methodology is applied to the incompressible Navier–Stokes equations and to linear as well as nonlinear elastic solid mechanics. The numerical examples presented feature convective effects and incompressible or nearly incompressible material. The numerical results, which are obtained with equal-order interpolation and without any stabilisation techniques, are smooth and accurate. It is shown that for p and h refinement, the theoretical rates of convergence are achieved. Copyright © 2014 John Wiley & Sons, Ltd. Received 19 March 2014; Revised 30 June 2014; Accepted 18 July 2014 KEY WORDS:
FEM; NURBS; isogeometric analysis; least-squares; Navier–Stokes; elasticity; incompressibility
1. INTRODUCTION This work presents a study of the Least-Squares FEM (LSFEM) based on NURBS. Comprehensive research and development of NURBS-based FEMs has been undertaken and widely published by Hughes and co-workers (see, for instance, [1, 2] and references therein). Such strategies are today commonly referred to as Isogeometric Analysis (IGA), and their applications include shell analysis, nearly incompressible solid mechanics, structural vibrations, structural optimisation, phase-transition phenomena, turbulent fluid dynamics and fluid-structure interaction (see, for instance, [3–10]). The primary motivation behind the development of IGA are the direct utilisation of geometry models employed in computer-aided design and the ease of constructing and implementing higher order basis functions. Many of the strategies developed in the context of Galerkin finite element methods (Galerkin-FEM) with Lagrangian basis functions have been reformulated in the isogeometric setting. This includes stabilisation techniques for advection dominated problems in fluid mechanics as well as strategies to circumvent or satisfy the LadyzhenskayaBabuLska-Brezzi (LBB) condition in incompressible fluid or solid materials (see, for instance, [2, 4, 5, 11, 12]). Least-squares finite element methods feature less prominently in literature than their Galerkin counterparts and are rarely used in the practice of computational fluid and solid mechanics. They either require a high degree of continuity of the basis functions or, if the field equations are rewritten as a lower order problem, they increase the number of solution variables. On the other hand, LSFEM
*Correspondence to: D. Peri´c, College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK. † E-mail:
[email protected] Copyright © 2014 John Wiley & Sons, Ltd.
´ C. KADAPA, W. G. DETTMER AND D. PERIC
strategies always yield symmetric and positive definite system matrices, even in the context of nonself-adjoint problems, and they are not subject to the LBB condition. As a result, they have been in the focus of a number of researchers in recent years. Bochev and Gunzburger have extensively investigated and developed the mathematical theory of LSFEM (see, for instance, [13–17], or [18] for a briefer introduction). The application of LSFEM to problems of fluid flow was investigated, for instance, in [19–28], while, in [29–38], LSFEM was used to solve problems in the area of linear and nonlinear elasticity as well as elasto-plasticity. In this work, a NURBS-based LSFEM is developed and applied to selected problems in fluid and solid mechanics. The purpose of this work is to investigate the potential of the proposed methodology. To the knowledge of the authors, NURBS-based LSFEM has not yet been reported in literature. The motivation for the implementation and study of NURBS-based LSFEM arises from three arguments: The application of LSFEM to a boundary value problem requires that the degree of continuity of the basis functions in the entire domain is at least equal to the degree of the highest derivative in the governing equation reduced by one. Thus, in the context of standard Lagrange basis functions, which are C 0 continuous across element boundaries, the governing equations need to be reformulated as a first-order problem by introducing auxiliary solution variables. This approach is commonly followed but is computationally expensive. The smoothness of the NURBS basis functions, even across inter element boundaries, allows, however, for the straightforward application of NURBS-based LSFEM to the original boundary value problems, which involve second or higher derivatives. Thus, the NURBS basis functions may be particularly suitable for LSFEM, and the properties of the resulting computational tool warrant careful investigation. Incompressible materials and advection-dominated problems should not require any special attention, that is, stabilisation techniques become superfluous and approximation spaces of different variables can be chosen independently. It is of great interest to investigate the potential of the NURBS-based LSFEM to overcome the problem of spurious oscillations. The system matrix is symmetric and positive definite. This is very desirable for large-scale simulations. At the same time, NURBS-based LSFEM may pose the following difficulties: The employment of a Newton–Raphson procedure to solve nonlinear problems is common practice in Galerkin-FEM. Typically, the consistent linearisation requires the first derivative of the weak form of the governing equations with respect to the discrete solution variables. In the context of LSFEM, however, it is necessary to provide the Hessian, that is, the second derivative of the strong form with respect to the discrete variables. This is tedious at best and often impractical. Thus, it is a common practice in nonlinear least-squares strategies to employ the Gauß–Newton procedure (see, for instance, [39, 40]). Here, the least-squares functional is formulated for the linearised strong form of the governing equations. Thus, the computation of the Hessian is avoided at the expense of slightly higher rates of convergence of the iteration. Galerkin type Isogeometric Analysis strategies allow for the employment of geometry models based on multiple patches. Across patch boundaries the solution is normally C 0 continuous. This is not admissible in the context of the proposed NURBS-based LSFEM due to the high requirements on continuity across the domain. Another important aspect of LSFEM is known as norm-equivalence. If the least-squares functional comprises several different residual terms associated with a set of governing equations and boundary conditions, the norms applied to each term cannot be chosen arbitrarily. In many standard cases, different norms are required to ensure norm-equivalence. An LSFEM that is not norm equivalent renders sub-optimal convergence of the solution. In this work, the (theoretical) requirement of norm-equivalence is ignored for the sake of simplicity of the methodology. The employment of the L2 -norm allows for the implementation of an element assembly procedure similar to standard Galerkin-FEM. Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
NURBS-BASED LSFEM FOR FLUID AND SOLID MECHANICS
If higher order problems are not rewritten as first order equation systems by means of auxiliary solution variables but, instead, sufficiently smooth approximation spaces are employed, then the conditioning of the system matrix deteriorates [17]. This may adversely affect practical computations. This work is organised as follows. In Sections 2 and 3, the fundamentals of LSFEM and of IGA and NURBS are revised, respectively. NURBS-based LSFEM is applied to incompressible Newtonian fluid flow in Section 4. In Section 5, the methodology is extended to linear and nonlinear elasticity. Sections 4 and 5 also include the presentation of a number of numerical examples. Conclusions are summarised in Section 5. 2. LEAST-SQUARES FINITE ELEMENT METHODS 2.1. Basic least-squares finite element method methodology This section, which closely follows [18], describes the fundamental idea behind LSFEM. Let be a bounded domain in Rd ; d D 1; 2 or 3, with a boundary . Let L be a linear differential operator and R be a boundary operator. Consider the problem given by Lu D f
in
and
Ru D g
on :
(1)
The least-squares functional is defined as the sum of the squares of appropriate norms of the residuals of the governing equations. Thus, it is always positive and convex. Throughout this work, the boundary conditions, including both Dirichlet and Neumann conditions, are applied in the leastsquares sense. The least-squares functional is then obtained as J .uI f; g/ D
1 kL u f k2H C kR u gk2H ; 2
(2)
where the operators k kH and k kH represent Hilbert space norms. Furthermore, let S denote a suitable Sobolev space, which contains the solution u. Then, the unconstrained minimisation problem min J .uI f; g/
(3)
u2S
is equivalent to (1). An LSFEM is based on restricting (3) to a finite element type subspace Sh S . Thus, the LSFEM approximation uh 2 Sh of the solution u 2 S is obtained by solving the problem min J .uh I f; g/ :
(4)
uh 2Sh
Accordingly, the first variation of the functional J .uh I f; g/ must vanish, which gives B .uh ; vh / D F .vh /
for all vh 2 Sh ;
(5)
where vh is the virtual counterpart of uh , and B .uh ; vh / D .L vh ; L uh /H C .R vh ; R uh /H
(6)
F .vh / D .L vh ; f /H C .R vh ; g/H :
(7)
On the basis of the discrete degrees of freedom uA , their trial counterparts vB and appropriate shape functions NA , with A; B D 1; 2; 3; : : : ; n, the approximations uh and vh may be expressed as uh D
n X AD1
Copyright © 2014 John Wiley & Sons, Ltd.
NA uA
and
vh D
n X
NB vB ;
(8)
BD1
Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
´ C. KADAPA, W. G. DETTMER AND D. PERIC
where n D dim.Sh /. Equation (5) then leads to the linear system of equations KuDf:
(9)
The elements of matrix K and vector f are obtained as KAB D .L NA ; L NB /H C .R NA ; R NB /H
(10)
fB D .L NB ; f /H C .R NB ; g/H :
(11)
Matrix K is symmetric and positive definite and therefore computationally convenient. If the least-squares functional is formulated in terms of the L2 -norm the system matrix K is sparse and can be assembled similarly to the stiffness matrix in Galerkin-FEM procedures. In particular, (2) and (10) become, respectively, Z Z 1 .L u f / .L u f / d C .R u g/ .R u g/ d J .uI f; g/ D (12) 2 Z KAB D
L NA L NB d C
Z
R NA R NB d :
(13)
In [17, 18], the LSFEM strategy outlined earlier is referred to as the ‘straightforward’ LSFEM. The major shortcoming consists in the fact that, in the context of standard finite element spaces based on Lagrange polynomials, it is applicable only to first-order problems. Other problems relate to norm equivalence and to the deterioration of the matrix condition number. Therefore, more sophisticated LSFEM has been developed (see [18] for an introduction). However, this work is restricted to the straightforward LSFEM because, as stated in Section 1, the objective is to investigate to what extent the employment of NURBS-based discretisation overcomes or lessens the shortcomings of the straightforward LSFEM. 2.2. Nonlinear problems In case of nonlinear problems, the least-squares functional can be constructed in terms of the nonlinear residuals of the governing equations. The standard Newton–Raphson procedure could be employed to find the solution of the resulting minimisation problem. However, the consistent linearisation requires the computation of the Hessian, that is, the second derivatives of the residuals with respect to the discrete solution vector u. This is tedious, computationally costly and often impractical. In order to avoid the need for the Hessian, the Gauß–Newton method may be employed. It is widely used in the fields of optimisation and nonlinear least-squares methods (see, for instance, [39, 40]). Instead of linearising the variation of the least-squares functional, the nonlinear residuals are linearised first, and a new least-squares functional is constructed in terms of linearised residuals. Thus, in each iteration of a load or time step, a linear least-squares problem is solved. Even though the Gauß–Newton method features slower convergence than the Newton procedure, it is often preferable as it does not require the costly computation of the Hessian of the residuals. In iteration step k C 1, the linearisation of a given nonlinear residual r D r.u; Q f; g/ is ˇ @ r.uI Q f; g/ ˇˇ .k/ .k/ Q I f; g/ C u ; (14) rN uI u ; f; g D r.u ˇ @u uDu.k/ ; f; g to formulate the least-squares functional where u.kC1/ D u.k/ C u. Using rN uh I u.k/ h .k/ JN uh I uh ; f; g in the subspace Sh leads to the minimisation problem min JN uh I u.k/ ; f; g : h
uh 2Sh
Copyright © 2014 John Wiley & Sons, Ltd.
(15)
Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
NURBS-BASED LSFEM FOR FLUID AND SOLID MECHANICS
The solution procedure for Equation (15) is exactly same as that for Equation (4) described above. The pseudo code for the Gauß–Newton procedure is similar to the standard Newton procedure and is given in Box 1.
3. NURBS AND ISOGEOMETRIC ANALYSIS 3.1. Geometry representation NURBS functions are standard tools in computer-aided design and computer graphics used to represent complex geometries. NURBS are a generalisation of B-splines. B-splines are parametric piecewise polynomial curves, which are composed of linear combinations of B-spline basis functions. Comprehensive information is provided in [1, 41–43]. For a given knot vector „ D ¹0 ; : : : ; nCaC1 º and degree of polynomial a, B-spline basis functions Ni;a can be defined recursively by ² 1 if i 6 6 i C1 Ni;0 ./ D 0 otherwise and Ni;a ./ D
i i CaC1 Ni;a1 ./ C Ni C1;a1 ./ i Ca i i CaC1 i C1
for a > 0 :
(16)
Given a control lattice consisting of control points B i;j;k with i D 0; 1; : : : ; n; j D 0; 1; : : : ; m; k D 0; 1; : : : ; l and the knot vectors „1 D ¹0 ; : : : ; nCaC1 º; „2 D ¹0 ; : : : ; mCbC1 º and „3 D ¹0 ; : : : ; lCcC1 º, a solid body can be described as a tensor product of univariate B-splines, that is, X .; ; / D
n X m X l X
Ni;a ./ Mj;b ./ Lk;c ./ B i;j;k
for
0 6 ; ; 6 1 :
(17)
i D0 j D0 kD0
Similarly, surfaces and curves are described on the basis of, respectively, control nets and control polygons. 3.2. NURBS spaces for isogeometric analysis In isogeometric analysis, NURBS basis functions are employed to approximate the field variables and to represent the geometry of the domain. Similarly to (17), a field variable u is approximated by uh .; ; / D
m X l n X X
Ni;a ./ Mj;b ./ Lk;c ./ ui;j;k
for 0 6 ; ; 6 1 :
(18)
i D0 j D0 kD0
Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
´ C. KADAPA, W. G. DETTMER AND D. PERIC
Figure 1. B-spline basis functions and knot vectors for a one-dimensional patch of four elements.
If a data structure is introduced such that every triple .i; j; k/ can be represented by an index A, Equation (18) may be written as
uh ./ D
nCmCl X
NA ./ uA ;
(19)
AD1
where uA represents the control variables similar to the control point coordinates. The vector contains the parametric coordinates ; and , while NA ./ denotes a coefficient in the tensor product of the univariate basis functions. For notational convenience, NURBS approximation spaces of order a are denoted by Qa in the remainder of this work. The continuity within a Qa patch can be of any order k with 0 < k < a, depending on the multiplicity of the internal knots. All examples presented in the remainder of this work feature a single patch without any multiple internal knots. The knot vectors considered in this work are open, that is, the approximation is interpolatory along the boundaries. Figure 1 shows examples of NURBS shape functions of different orders for a one-dimensional patch of four elements. All basis functions shown are C a1 continuous because there are no multiple internal knots. It can be observed that basis functions are interpolatory at the boundary.
4. FLUID MECHANICS 4.1. The incompressible Navier–Stokes 4.1.1. Governing equations. Let be a region in Rd and the boundary of , where d denotes the number of spatial dimensions. The steady-state Navier–Stokes equations may then be written as .ru/ u C rp r 2 u D f
r uD 0
uD g
in
on
in
(20)
(21)
(22)
where ; u; p and denote, respectively, the fluid density, velocity, pressure and viscosity. The quantities f and g represent, respectively, body forces and velocity boundary values. Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
NURBS-BASED LSFEM FOR FLUID AND SOLID MECHANICS
4.1.2. Least-squares finite element formulation. As discussed in Section 2, the employment of the Gauß–Newton method requires the linearisation of the governing equations. At iteration step .kC1/, the linearisation of Equations (20)–(22) yields ru.k/ uC.ru/u.k/ Crp r 2 u D f ru.k/ u.k/ rp .k/ C r 2 u.k/ (23) r u D r u.k/
(24)
u D g u.k/ :
(25)
The expressions on the left hand side of Equations (23)–(25) are linear in terms of u and p, while the expressions on the right hand side are constant. The least-squares functional based on the L2 norm is obtained as 1 JN u; pI u.k/ ; p .k/ ; f ; g D ru.k/ u C .ru/u.k/ C rp r 2 u 2 2 f ru.k/ u.k/ rp .k/ C r 2 u.k/ 2 (26) L
2 2 C r u C r u.k/ 2 C u g u.k/ 2 L
L
By introducing a NURBS-based discretisation of the domain , the velocity and the pressure can be approximated by uh D Nu u
ph D Np p ;
and
(27)
where a more compact notation than in Equation (8) has been used and u and p denote the degrees of freedom associated with the control points, while the matrices Nu and Np represent the basis functions for the discretisation of the velocity and the pressure, respectively. Similar approximations are used for u.k/ ; p .k/ ; u; p and the test functions v; q , that is, D Nu u.k/ ; u.k/ h
ph.k/ D Np p.k/
uh D Nu u ;
ph D Np p
v h D Nu v ;
qh D Np q :
(28)
By using Equations (28) in the first variation of the least-squares functional and equating it to zero, one obtains the linear system of equations K uQ D f :
(29)
where uQ D ¹u; pºT is the vector of unknowns. The matrix K and the vector f are computed as explained in Section 2. It is noted that the treatment of Neumann boundary conditions including the standard outflow boundary condition in the context of the proposed NURBS-based LSFEM is straightforward. However, the numerical examples presented in the next subsection feature only Dirichlet boundaries. Thus, for the sake of brevity, the Neumann boundary condition has not been considered here. It is included in Section 5. 4.2. Numerical examples In the following, NURBS shape functions of degree a are denoted by Qa and used equally for the velocity and the pressure field. A direct solver is employed for the solution of the resulting linear equation systems. 4.2.1. Kovasznay flow. In order to study the performance of the proposed formulation, the so-called Kovasznay flow is considered [44]. The problem is two dimensional and stationary and governed by Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
´ C. KADAPA, W. G. DETTMER AND D. PERIC
the incompressible Navier–Stokes equations. The domain is D Œ0:5; 1:5 Œ0:5; 1:5 and the analytical solution is given by u.x; y/ D 1:0 e x cos.2 y/ v.x; y/ D
(30)
x e sin.2 y/ 2
(31)
1 p.x; y/ D p0 e 2x ; 2 where Re D 2
r
(32)
Re 2 C 4 2 : 4
(33)
The quantities Re and p0 represent, respectively, the Reynolds number and a reference pressure. For Re D 40, the convergence of the velocity and the pressure is studied as the discretisation is refined with p-refinement and h-refinement. The meshes considered are uniform, that is, they possess equal knot spans in both directions. The rates of convergence obtained are shown in Figure 2. For each graph, that is, for each degree a of the NURBS basis functions, the obtained approximate convergence ratio r is given. It is observed that, for the L2 norm of the velocity components, r > a C 1 for a > 3 while, for the L2 norm of
Figure 2. Kovasznay flow: error norms for Kovasznay flow with Re D 40. Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
NURBS-BASED LSFEM FOR FLUID AND SOLID MECHANICS
the pressure and the H1 norm of the velocity, r > a for a > 2. Thus, convergence is poor for lower order basis functions, whereas it is optimal for quartic or higher order NURBS discretisations. This behaviour is due to the use of the formulation which does not satisfy the strict requirement of normequivalence in the least-squares functional. It emphasises the benefit of the employment of smooth high-order NURBS shape functions in the context of least-squares finite elements. Table I shows the convergence of the norm of the right hand side at each iteration of the Gauß– Newton procedure for different mesh densities with quartic .Q4 / NURBS. It is observed that the convergence improves as the mesh is refined, and the asymptotically quadratic rate of convergence of the Newton–Raphson method is recovered. Contour plots of the velocity components and the pressure as well as streamline plots are shown in Figure 3. The diagrams obtained with a > 3 are indistinguishable from each other. The pressure and velocity fields do not exhibit any spurious oscillations, which confirms the assertion that the proposed strategy does not require the employment of different approximation spaces or a stabilisation technique in order to model fluid incompressibility or convection effects. 4.2.2. Flow in lid-driven cavity. This is a benchmark problem for numerical methods for incompressible viscous fluid flow. The geometry and boundary conditions are as shown in Figure 4(a). The fluid velocity is zero on all the boundaries except at the lid where the horizontal velocity component is non-zero and uniform. The velocity at the top corners is equal to that of the lid, thus making it a ‘leaking cavity’. The non-uniform mesh shown in Figure 4(b) is employed, which features finer discretisation in the corners of the domain. Different mesh densities are used to study the flow at different Reynolds numbers. Meshes of 20 20; 40 40 and 150 150 are used to study flows with Reynolds number 100, 1000 and 10 000, respectively. The velocity profiles obtained with the proposed methodology are compared with the widely accepted results obtained by Ghia et al [45]. As shown in Figure 5, they agree accurately with the benchmark values. The results obtained for basis functions with order a > 3 are indistinguishable. For Re D 10 000, the solution failed to converge for quadratic NURBS. Some contour and streamline plots are presented in Figure 6. All of the pressure and velocity fields obtained are free from spurious oscillations. 5. SOLID MECHANICS 5.1. Nearly incompressible linear elasticity 5.1.1. Governing equations. The governing equations for a linear elastic body read r Df uDg nDt
on
in ;
(34)
D ;
(35)
in N ;
(36)
Table I. Kovasznay flow: evolution of the norm of the right hand side with Q4 NURBS. Norm of right hand side Iteration 1 2 3 4 5 6 7 8 9
5 5 mesh
10 10 mesh
20 20 mesh
2.5010 EC00 7.3951 EC00 6.8924 E01 2.3512 E02 4.0363 E04 4.0116 E05 4.7320 E06 5.7093 E07 7.5774 E08
2.1437 EC00 7.7189 EC00 1.1707 EC00 2.9692 E02 1.4740 E05 1.5498 E08
1.8358 EC00 7.8856 EC00 8.3107 E01 2.0521 E02 1.3929 E05 3.7139 E11
Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
´ C. KADAPA, W. G. DETTMER AND D. PERIC
Figure 3. Kovasznay flow: velocity and pressure contour plots for 20 20 mesh with Q2 and Q3 NURBS.
where D D [ N is the boundary of the body . The vector fields u; g; f ; t and n represent, respectively, the displacement, the prescribed displacement on D , the body force, the prescribed traction on N and the outward unit normal on . The stress tensor is defined as D 2 " C tr."/ ; Copyright © 2014 John Wiley & Sons, Ltd.
(37) Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
NURBS-BASED LSFEM FOR FLUID AND SOLID MECHANICS
Figure 4. Lid-driven cavity: geometry, boundary conditions (a) and a typical mesh used for the analysis (b).
where and are the Lamé parameters, and the strain tensor " is related to the displacements by "D
1 ru C ruT : 2
(38)
By introducing the pressure p and the deviatoric strain tensor "dev as, respectively, pD
1 tr. / ; 3
and
"dev D "
1 tr."/ I 3
(39)
the stress tensor can be rewritten as D 2 "dev C p I :
(40)
The pressure, expressed in terms of the displacement, becomes p D r u;
(41)
2 is the bulk modulus. where D C 3 Thus, the governing equations for linear elasticity can be written in terms of the displacement and the pressure as u C ˛ rp D f
r uD
uD g
nD t where, ˛ D 1 C
3
p
in
in
(42)
(43)
on D
on N
(44)
(45)
and the stress is given by Equation (40).
Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
´ C. KADAPA, W. G. DETTMER AND D. PERIC
Figure 5. Lid-driven cavity: velocity profiles along the horizontal and vertical centre lines for 150 150 mesh.
5.1.2. Least-squares finite element formulation. The least-squares functional based on the L2 norms of the residuals of (42)–(45) is given by
J .u; pI f ; g/ D
1 2
p 2 r 2 u C ˛ rp f 2 2 C r u L
L2 : C k u g k2L2 C k n t k2L2 D
Copyright © 2014 John Wiley & Sons, Ltd.
(46)
N
Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
NURBS-BASED LSFEM FOR FLUID AND SOLID MECHANICS
Figure 6. Lid driven cavity flow: streamlines and pressure contour lines obtained with Q4 NURBS.
By introducing a NURBS-based discretisation of the domain , the displacement and the pressure can be approximated by uh D Nu u
and
ph D Np p ;
(47)
where u and p denote the degrees of freedom associated with the control points, while the matrices Nu and Np represent the basis functions for the discretisation of the displacement and the pressure, respectively. Similar approximations are used for the test functions v and q of u and p, that is, vh D Nu v
and
qh D N p q :
(48)
By using Equations (47) and (48) in the first variation of the least-squares functional (46) and equating it to zero, one obtains the linear system of equations K uQ D f ; Copyright © 2014 John Wiley & Sons, Ltd.
(49) Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
´ C. KADAPA, W. G. DETTMER AND D. PERIC
where uQ D ¹u; pºT is the vector of unknowns. The matrix K and the vector f are computed as discussed in Section 2. 5.2. Nearly incompressible nonlinear elasticity 5.2.1. Governing equations. In the regime of finite deformations, the governing equations for an elastic body may be expressed in terms associated with the deformed configuration as rx D f
in
uDg
on D ;
nDt
;
(50) (51)
on N ;
(52)
where and its boundary D D [N represent the deformed configuration. The tensor denotes the Cauchy stress, while f and t represent the body force and the traction force in the deformed configuration. Here, rx is the gradient operator with respect to spatial coordinates. Preliminary to the definition of the stress, the deformation gradient F , the right Cauchy-Green tensor C and the left Cauchy-Green tensor B are introduced as @u (53) F DIC @X C D F TF
(54)
B D FFT ;
(55)
where X represent the coordinates in the undeformed configuration. The isochoric part of the right Cauchy-Green tensor can expressed as 2 CN D J 3 C
J D det.F / :
with
(56)
For the formulation of nearly incompressible elastic material, it is convenient to employ a free energy function which can be split additively into an isochoric and a volumetric component, that is, W .CN ; J / D W iso .CN / C W vol .J / : Here, a Neo-Hooke type free energy is chosen and 1 1 W .CN ; J / D tr.CN / 3 C 2 2
(57)
1 2 .J 1/ lnJ 2
:
(58)
From (58), the Cauchy stress is obtained by the standard procedure as D dev C p I ;
(59)
where
dev
D J
5 3
1 B tr.B/ I 3
pD 2
and
1 J J
:
(60)
Details on hyperelastic solid mechanics may be found, for instance, in [46, 47]. In terms of the displacement and the pressure, the governing equations for nearly incompressible Neo-Hooke elastic materials can then be written in the deformed configuration as rx dev C r p D f J
1 D J
uD g
2p
in on D
dev C p I n D t
Copyright © 2014 John Wiley & Sons, Ltd.
in
on N
(61) (62) (63) (64) Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
NURBS-BASED LSFEM FOR FLUID AND SOLID MECHANICS
5.2.2. Least-squares finite element formulation. The Gauß–Newton method, which is employed to resolve the nonlinearity of the problem, requires that the least-squares functional is formulated in terms of the linearised residuals of the governing equations. The linearisation of the deviatoric part of the Cauchy stress may be written as @ dev .u/ u … „ @uƒ‚ dev C ;
dev .u C u/ D dev .u/ C D dev .u/
(65) (66)
where
dev
5 5 D .rx u/ dev C J 3 3
2 T B .rx u/ C rx u B .B W rx u/ I : (67) 3
From (61)–(67), the set of linearised governing equations is then obtained as rx dev C p I D f rx dev .u.k/ / C p .k/ I 2 p .k/ 1 2 1 .k/ .k/ J C .k/ rx u p D J .k/ C
J J u D g u.k/
in
in
on D
(68)
(69)
(70)
dev C p I n D t dev .u.k/ / C p .k/ I n on N :
(71)
It is noted that, for convenience and at the expense of slightly higher rates of convergence, the dependency of the current domain configuration and of the vector fields f ; t and n on the displacement (68)–(71), the is not linearised in this work. Based on the L2 -norm of the residuals of Equations formulation of the least-squares functional JN u; pI u.k/ ; p .k/ ; f ; g; t is straightforward and omitted here for the sake of brevity. Using NURBS-based approximations similar to Equation (28) for the displacement, the pressure, their increments and their virtual counterparts yields the linear system K uQ D f ;
(72)
where uQ D ¹u; pºT contains the unknown increments of the control point variables. 5.3. Numerical examples Similar to the examples on fluid flow, the following test cases employ equal order interpolation for the displacement and the pressure fields. A direct solver is employed for the solution of the resulting linear equation systems. 5.3.1. Cook’s membrane - linear elasticity. Cook’s membrane is one of the benchmark problems used to assess the performance of finite element formulation for incompressible or nearly incompressible solids (see, for instance, [48, 49]). Studies based on IGA are presented in [4, 5, 11, 50]. The geometry and the boundary conditions of the problem are shown in Figure 7(a). Young’s modulus is E D 240:565 MPa, and Poisson’s ratio is D 0:4999. The load F D 100 N/mm is uniformly distributed along the vertical edge, and the problem is run under plane strain conditions. Five different discretisations, that is, 1 1, 2 2, 4 4, 8 8 and 16 16 elements, are employed. The 8 8 mesh is shown in Figure 7(b). Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
´ C. KADAPA, W. G. DETTMER AND D. PERIC
The diagram in Figure 8 demonstrates the convergence of the vertical displacement of the top right corner (Point A) as the discretisation is refined. As shown, the results obtained with the current LSFEM are in excellent agreement with those obtained with two-field mixed Galerkin formulations for NURBS, studied in [5, 50]. The accuracy increases substantially as the order of the discretisation is increased from quadratic to cubic. Cubic and higher order discretisations render accurate results
Figure 7. Cook’s membrane: geometry with boundary conditions (a) and 8 8 mesh (b).
Figure 8. Cook’s membrane (linear): convergence for different orders of NURBS basis functions.
Figure 9. Cook’s membrane (linear): contour plots of results for 16 16 mesh with Q3 NURBS. Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
NURBS-BASED LSFEM FOR FLUID AND SOLID MECHANICS
Figure 10. Cook’s membrane (nonlinear): convergence for different orders of NURBS basis functions.
Figure 11. Cook’s membrane (nonlinear): contour plots of results for 16 16 mesh with Q3 NURBS.
for relatively coarse meshes and show a uniform convergence pattern towards the reference solution. For Q2 NURBS and for higher order NURBS with coarse meshes mixed Galerkin formulation gives better results compared with LSFEM. However, for denser meshes with higher order NURBS .Qa ; a > 2/; the results obtained with LSFEM are indistinguishable from those obtained in [5, 50]. Figure 9 shows the contour plots of the pressure and the von-Mises stress for 16 16 mesh. The plots are smooth and the singularity in the top left corner is captured well. 5.3.2. Cook’s membrane - nonlinear elasticity. The geometry, the loading and the boundary conditions of the problem are identical to Section 5.3.2. Nearly incompressible Neo-Hookean material as defined by Equation (58) is considered and the uniformly distributed load F is conservative. The convergence of the solution and typical contour plots of the pressure and the von-Mises stress are displayed, respectively, in Figures 10 and 11. The quality of the results is similar to the linear case discussed in Section 5.3.2. The converged solution agrees with [4, 5, 50].
6. CONCLUSIONS A generic NURBS-based version of the LSFEM has been presented and applied to the Navier– Stokes equations for incompressible fluid flow and to linear as well as nonlinear elastic solid mechanics. The least-squares functional is formulated straightforwardly in terms of the governing equations and boundary conditions. The reduction to first order problems by means of auxiliary Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
´ C. KADAPA, W. G. DETTMER AND D. PERIC
variables is not required. The proposed methodology has been applied to the solution of several standard benchmark problems. The results obtained for the test cases are very promising and, for cubic or higher order NURBS basis functions, the accuracy is excellent. On the basis of equal order interpolation for the velocity and the pressure fields, the proposed methodology does not require special considerations such as stabilisation techniques. The numerical results are smooth and completely free of any spurious oscillations despite the presence of strong advection and incompressibility effects. The relatively poor performance of lower order basis functions is attributed to the use of the formulation, which does not satisfy the strict requirement of norm-equivalence. The Gauß–Newton procedure is robust. Asymptotically quadratic rates of convergence are recovered as the discretisation is refined. The direct solver employed for the solution of the linear systems has not encountered any difficulties due to poor condition numbers. It is possible that this issue may require some attention in the context of iterative solvers. Summarising, it has been demonstrated in this paper that the employment of NURBS basis functions in the context of LSFEM is viable and offers an alternative to Galerkin-FEM. The outstanding feature of the proposed methodology is clearly the simplicity of the formulation and its applicability to a wide range of physical problems. REFERENCES 1. Cottrell JA, Bazilevs Y, Hughes TJR. Isogeometric Analysis: Toward Integration of CAD and FEA. John Wiley & Sons: Chichester, England, 2009. 2. Hughes TJR, Bazilevs Y, Cottrell JA. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering 2005; 194:4135–4195. 3. Cirak F, Ortiz M, Schröder P. Subdivision surfaces: a new paradigm for thin-shell finite-element analysis. International Journal for Numerical Methods in Engineering 2000; 47(12):2039–2072. ¯ and F¯ projection methods for nearly incompressible linear and 4. Elguedj T, Bazilevs Y, Calo VM, Hughes TJR. B non-linear elasticity and plasticity using higher-order NURBS elements. Computer Methods in Applied Mechanics and Engineering 2008; 197:2732–2762. 5. Kadapa C, Dettmer WG, Peri´c D. Mixed methods for isogeometric analysis of nearly incompressible materials. submitted for publication. 6. Cottrell JA, Reali A, Bazilevs Y, Hughes TJR. Isogeometric analysis of structural vibrations. Computer Methods in Applied Mechanics and Engineering 2006; 195:5257–5296. 7. Wall WA, Frenzel MA, Cyron C. Isogeometric structural shape optimization. Computer Methods in Applied Mechanics and Engineering 2008; 197:2976–2988. 8. Gomez H, Hughes TJR, Nogueira X, Calo VM. Isogeometric analysis of the isothermal Navier–Stokes–Korteweg equations. Computer Methods in Applied Mechanics and Engineering 2010; 199:1828–1840. 9. Bazilevs Y, Michler C, Calo VM, Hughes TJR. Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes. Computer Methods in Applied Mechanics and Engineering 2010; 199:780–790. 10. Bazilevs Y, Calo VM, Hughes TJR, Zhang Y. Isogeometric fluid-structure interaction: theory, algorithms and computations. Computational Mechanics 2008; 43:3–37. 11. Mathisen KM, Okstad KM, Kvamsdal T, Raknes SB. Isogeometric analysis of finite deformation nearly incompressible solids. Journal of Structural Mechanics 2011; 44(3):260–278. 12. Taylor RL. Isogeometric analysis of nearly incompressible solids. International Journal for Numerical Methods in Engineering 2011; 87:273–288. 13. Bochev P, Gunzburger M. A least-squares finite element method for the Navier-Stokes equations. Applied Mathematics Letters 1993; 6:27–30. 14. Bochev P, Gunzburger M. Accuracy of least-squares methods for the Navier-Stokes equations. Computers & Fluids 1993; 22:549–563. 15. Bochev P, Gunzburger M. Analysis of least-squares finite element methods for the Stokes equations. Mathematics of Computation 1994; 63:479–506. 16. Bochev P. Analysis of least-squares finite element methods for the Navier-Stokes equations. SIAM Journal on Numerical Analysis 1997; 34:1817–1844. 17. Bochev P, Gunzburger M. Least-Squares Finite Element Methods. Springer: New York, USA, 2009. 18. Bochev P, Gunzburger M. Least-squares finite element methods. Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006; 1137–1162. 19. Jiang BN, Carey GF. A stable least-squares finite element method for non-linear hyperbolic problems. International Journal for Numerical Methods in Fluids 1988; 8(8):933–942. 20. Jiang BN, Carey GF. Least-squares finite element methods for compressible Euler equations. International Journal for Numerical Methods in Fluids 1990; 10(5):557–568. Copyright © 2014 John Wiley & Sons, Ltd.
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Int. J. Numer. Meth. Engng (2014) DOI: 10.1002/nme
