Solving thermal and phase change problems with the eXtended finite ...

Computational Mechanics 28 (2002) 339–350 Ó Springer-Verlag 2002 DOI 10.1007/s00466-002-0298-y

Solving thermal and phase change problems with the eXtended finite element method R. Merle, J. Dolbow

339 Abstract The application of the eXtended finite element method (X-FEM) to thermal problems with moving heat sources and phase boundaries is presented. Of particular interest is the ability of the method to capture the highly localized, transient solution in the vicinity of a heat source or material interface. This is effected through the use of a time-dependent basis formed from the union of traditional shape functions with a set of evolving enrichment functions. The enrichment is constructed through the partition of unity framework, so that the system of equations remains sparse and the resulting approximation is conforming. In this manner, local solutions and arbitrary discontinuities that cannot be represented by the standard shape functions are captured with the enrichment functions. A standard time-projection algorithm is employed to account for the time-dependence of the enrichment, and an iterative strategy is adopted to satisfy local interface conditions. The separation of the approximation into classical shape functions that remain fixed in time and the evolving enrichment leads to a very efficient solution strategy. The robustness and utility of the method is demonstrated with several benchmark problems involving moving heat sources and phase transformations. Keywords Partition-of-unity, Phase transformation, Finite element, X-FEM

1 Introduction A number of thermal and thermo-mechanical problems of interest to the engineering and materials science communities are characterized by the presence of a highly localized, moving heat source. Examples include the heat dissipated near a high-speed crack front (Li et al. 1996), aerothermal loads on engine structures during hypersonic flight (Wieting et al. 1991), and thermal shock waves (Tzou 1992). Similarly, the motion of phase boundaries is often accompanied by a release of latent heat. The material interfaces represent surfaces across which several fields Received 20 May 2001 / Accepted 19 December 2001

R. Merle Department of Mechanical Engineering, Ecole Normale Superieure de Cachan, France J. Dolbow (&) Department of Civil and Environmental Engineering, Duke University, Box 902 87 Durham, North Carolina 27708-0287 e-mail: [email protected]

may be discontinuous, and the evolution of phase boundaries is often governed by local conditions such as kinetic relations in shape memory alloys (Abeyaratne et al. 1994), or the classical Gibbs–Thomson equations for the dendritic solidification of pure substances. Numerical methods provide an important resource for studying these classes of problems, especially when the salient features evolve in space–time windows that are only accessible to sophisticated experimental techniques. Most finite element models for these classes of problems have employed either adaptive remeshing or moving-mesh algorithms. Tamma and Saw (1989) developed an adaptive p-version finite element approach to refine solutions locally. Probert et al. (1992) utilized an automatic triangular mesh generator in conjunction with a simple error estimator to model heat conduction problems with moving heat sources. A central consideration in phase transformation models that assume a sharp interface is the need to capture discontinuities in temperature gradients. A moving mesh for phase transformation problems originally developed by Lynch and O’Neill (1981) has been used extensively and extended to directional solidification problems in three dimensions (Sampath and Zabaras 1999). Another moving-mesh method along these lines was developed by Kuang and Atluri (1985). An alternative ALE approach to the same class of problems was developed by Ghosh and Moorthy (1993). The latter article also provides a review of the wide variety of numerical methods employed to solve diffusive phase change problems. A new approach for representing localized behavior has recently emerged in the finite element literature, termed the partition of unity method (Melenk and Babusˇka 1996). The main idea is to extend a classical approximation by augmenting the set of nodal shape functions with products of a subset of these same shape functions and local enrichment functions. The enrichment functions may be selected using either specific information about the local solution (i.e. order of singularity) or general information (i.e. degree of continuity). The eXtended finite element method (X-FEM) (Dobow 1999) is a variation on this framework that seeks to model arbitrarily evolving geometric features (such as cracks and interfaces) with evolving enrichment functions. It has been applied to the modeling of arbitrary crack growth without remeshing (Moe¨s et al. 1999), and toward examining the effects of microstructure on stress intensity factors in functionally graded materials (Dolbow and Nadeau 2001). In the present paper, the application of the X-FEM to thermal and nonlinear phase change problems is

340

developed. Particular attention is drawn to the time-dependence of the basis, and an alternative formulation of a standard time-stepping algorithm is presented. Details of the construction of the enrichment functions as well as the iterative procedure used to satisfy the nonlinear constitutive law at the phase boundary are discussed. Through the examination of several example problems, we demonstrate how the present method provides for excellent accuracy in the vicinity of moving heat sources and phase boundaries without recourse to remeshing. This paper is organized as follows. In the next section, we provide the problem formulation for the general case of transient thermal analysis with an embedded phase Fig. 2. Zoom of the interface CI boundary. Section 3 outlines the discretization with the X-FEM, weak formulation, and time-stepping algorithm. The results from several numerical examples are then with Tg and q
prescribed temperatures and heat fluxes, provided in Sect. 4. Finally, Sect. 5 provides a summary respectively. and some concluding remarks. The evolution of the surface CI is described by a Stefan condition, written as:

2 Problem formulation We begin by considering the domain X which is divided into the regions X1 and X2 by the phase interface CI as shown in Fig. 1. A zoom of the interface is shown in Fig. 2. The special case of a domain occupied by a single region without a phase boundary is easily recovered. Knowing the initial temperature in the domain, Tðx; 0Þ, we seek to determine the evolution of the function Tðx; tÞ which satisfies the governing equations, boundary conditions, and interfacial conditions. The governing equations are given by: c1 T;t ¼ r  ðj1  rTÞ þ f c2 T;t ¼ r  ðj2  rTÞ þ f

in X1 (solid phase) in X2 (liquid phase)

ð1aÞ ð1bÞ

q
ð3Þ L where L is the volumetric latent heat of fusion, and Vn is the normal velocity at a point xI on the interface. The normal vector nI is assumed to point into the liquid phase. The equation arises by considering the localization of the conservation of energy onto the surface of discontinuity CI , and involves the jump in interfacial heat flux Vn ðxI Þ ¼

q
¼ ðj1  rTÞjCI  nI  ðj2  rTÞjCþ  nI I

ð4Þ

Additional constraints may be prescribed on the phase boundary, which we write in the general form:

GðT; Vn ; GÞ ¼ 0 on CI

ð5Þ

where c1 ; c2 are the volumetric heat capacities of the phases, and j1 ; j2 are the thermal conductivity tensors. In the above, f denotes a heat source that may evolve in both space and time. The above are supplemented with external boundary conditions and an initial condition

where G consists of geometric quantities such as curvature and normal orientation. Examples include the classical Gibbs–Thomson relations for the unstable dendritic growth of crystals into an undercooled melt. In the present investigation, we consider the simple case given by

T ¼ Tg on Cg ðj  rTÞ  n ¼ q
on Ch Tðx; 0Þ ¼ T0 ðxÞ

GðT; Vn ; GÞ ¼ T  Tm

ð2aÞ ð2bÞ ð2cÞ

on

CI

ð6Þ

where Tm is the melting temperature. This completes the description of the boundary value problem we wish to solve.

3 Discretization with the X-FEM We now describe the discrete formulation. Considering that the motion of the interface will be described by enrichment functions, there are three issues we need to be concerned with: the choice of enrichment functions; the time-stepping algorithm; the constraint given by Eq. (6).

Fig. 1. Domain split into X1 and X2 by the surface CI

We note that the last two issues follow from the first. In other words, modifications to standard time-stepping and constraint strategies are required due to the nature of the enriched approximation. These are addressed in the following subsections.

3.1 The enriched approximation A standard Galerkin approximation begins by considering a finite dimensional subspace Uh0 spanned by N linearly independent functions of the solution space U0 . For the sake of concreteness, we now consider a rectangular domain X and a regular finite element triangulation h Th ¼ [nel e¼1 Te such that T ¼ X as shown in Fig. 2a. This figure also depicts an interface whose geometry CI is taken to be independent of the mesh. A standard finite element basis for Uh0 is constructed from the space of complete polynomials Pk ðThe Þ of order  k over each element: Uh0 ¼ spanf/i gNi¼1 where f/i 2 ½C0 ðTh Þ2 : /i jThe 2 ½Pk ðThe Þ2 and /i jCg ¼ 0g : ð7Þ The functions /i ðxÞ are typically the nodal shape functions. Any linear combination of these functions results in a continuous interpolation for the temperature field, and furthermore possesses poor approximation properties for representing arbitrary discontinuities. Standard finite element approaches therefore construct moving meshes that conform to the interface geometry in order to represent the discontinuity, and employ significant mesh refinement near moving heat sources. The X-FEM takes an alternative approach by extending the standard finite element approximation. We consider the set of overlapping subdomains fxi g defining the support of each nodal shape function (shown in Fig. 3b) and sets of enrichment functions fEki g that possess desirable approximation properties over each subdomain. The method follows the partition-of-unity framework (Melenk and Babusˇka 1996) through multiplying the enrichment functions by the nodal shape functions /i in order to ensure a conforming approximation. A general X-FEM basis for Uh0 is therefore

The approximation to the temperature field TðxÞ is then written as

TðxÞ ¼

X

Ti /i ðxÞ þ

X

i

/i ðxÞ

i

X

! aki Eki ðxÞ

ð9Þ

k

ðkÞ

where the Ti and ai are the constant degrees of freedom. It bears emphasis that as a consequence of the above construction, the coefficients Ti are not equal to the value of the temperature field T h ðxi Þ at the nodes. The above construct is the most general form of the X-FEM approximation where every nodal shape function may be enriched with an arbitrary number of additional functions fEki g. In practice, only those functions whose supports are in the vicinity of a feature of interest are enriched, giving the approximation a local character. For example, the X-FEM approximation for a phase interface is given by

X

T h ðxÞ ¼

Ti /i ðxÞ

þ

i2I

|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

classical approximation

X

bj /j ðxÞgðxÞ

ð10Þ

j2J

|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} enrichment

where I is the set of all nodes in the mesh, J is the set of nodes that form a partition of unity for the function gðxÞ. We define this set as

J ¼ fj 2 I : xj \ CI 6¼ ;g

and use N E to denote the size of the set. As shown in Fig. 3, only those nodes in the vicinity of the interface are selected; the X-FEM extends a standard finite element approximation locally. We now focus attention on the form of the enrichment function gðxÞ. This function should be chosen to provide (10) with ‘‘good’’ approximation properties in the vicinity of the phase interface. For certain problems, the form of the local solution can be anticipated in advance; examples n oN E are provided in Sect. 4. In these cases, gðxÞ can be chosen Ni Uh0 ¼ span f/i g [ f/i Eki gk¼1 ð8Þ by identifying the part of the solution that cannot be welli¼1 represented by polynomials or that is moving and highly where N is the number of standard nodal shape functions localized. and NiE is the number of enrichment functions for node i. In the case of phase transformations, fairly little information may be known about the local solution. At a minimum, however, we expect a discontinuity in temperature gradient normal to the interface as specified by (4). A classical approximation to the temperature field has a gradient given by

rT h ðxÞ ¼

X

Ti r/i ðxÞ

i2I

With the standard C 0 ðXÞ shape functions, the above contains jumps only across element boundaries. This is precisely the consideration that motivates moving-mesh strategies that match element boundaries to the phase interface. By way of contrast, (10) has the gradient

rT h ðxÞ ¼ Fig. 3. a An arbitrary phase interface placed on a mesh. The shape functions of the squared nodes form a partition of unity for enrichment. b The support of a single nodal shape function

X

Ti r/i ðxÞ þ

i2I

þ /j ðxÞrgðxÞÞ

X

bj ðr/j ðxÞgðxÞ

j2J

ð11Þ

341

342 Fig. 4. A piecewise linear function on an element interior and its projection onto a classical finite element space

so that arbitrary gradient jumps can be represented by a suitable choice of the function gðxÞ. This concept was initially suggested in the context of meshfree methods by Krongauz and Belytschko (1998). We now consider a simple one-dimensional example to obtain such a function. Figure 4 shows a piecewise-linear function on an element interior. For the sake of simplicity, we illustrate the main idea in local element coordinates n 2 ½0; 1, such that /1 ¼ 1  n and /2 ¼ n. The approximation on the element is therefore

T h ðnÞ ¼ ð1  nÞ  ðT1 þ gðnÞb1 Þ þ n  ðT2 þ gðnÞa2 Þ The projection of this function onto a space of classical shape functions is also shown in the figure. Clearly, the classical shape functions alone are not able to represent the derivative discontinuity at xf . We may in fact choose gðxÞ to be piecewise-linear, and it is easy enough to show that if T1 ¼ T2 ¼ 0 and a1 ¼ a2 ¼ 1:0 then gðxÞ ¼ T h ðxÞ. The problem with using a ‘‘hat’’-like function for gðxÞ is that such a function may not be linearly independent of the classical shape functions, especially when the phase boundary is at a node. Perhaps more importantly, a recent study (Ji 2001) indicates that the use of such functions may not lead to convergence in local quantities such as the gradient discontinuity at the phase boundary. The consequence of using such functions are examined in the numerical examples in Sect. 4. The simplest enrichment function that can represent the discontinuous derivative shown in Fig. 4 turns out to be a Heaviside function. We consider the enrichment function

3.2 Variational formulation and time-stepping The usual approach to implementing a time-integration scheme with finite elements is to discretize in time after developing the weak formulation in space. For reasons to be described later, we will switch the order of these steps. Henceforth, we will also limit our derivation to onedimensional problems. We consider the solution on the time interval ½0; tf , partitioned into time steps as ½t n ; t nþ1 . Considering the thermal conductivities to be isotropic and constant, we write (1) at time step t nþ1 as: nþ1 c1 T;tnþ1 ¼ j1 T;nþ1 xx þ f

on

X1

ð12aÞ

c2 T;tnþ1

on

X2

ð12bÞ

¼

j2 T;nþ1 xx

þf

nþ1

and we consider the generalized trapezoidal time-stepping algorithm characterized by the parameter a:



T nþ1 ¼ T n þ Dt aT;nþ1 þ ð1  aÞT;nt t

ð13Þ

Substituting this equation into the above yields, after rearranging some terms:

c1 T nþ1  aDtj1 T;nþ1 ¼ c1 ðT n þ ð1  aÞDtT;tn Þ þ f nþ1 x c2 T nþ1  aDtj2 T;nþ1 ¼ c2 ðT n þ ð1  aÞDtT;tn Þ þ f nþ1 x We now multiply by a weight function dT nþ1 , and integrate over the domain at time step n þ 1. After performing the usual integration by parts, this gives:

ðdT nþ1 ; c1 T nþ1 ÞXnþ1 þ aDtBðdT nþ1 ; T nþ1 Þ 1

¼ ðdT nþ1 ; f nþ1 ÞXnþ1 þ ðdT nþ1 ; c1 T n ÞXnþ1 1

1

þ ð1  aÞDtðdT nþ1 ; c1 T;nt ÞXnþ1 1

þ aDtðdT ðdT

nþ1

; c2 T

¼ ðdT

nþ1

nþ1

nþ1

; j1 T;xnþ1 nI ÞCI

ÞXnþ1 þ aDtBðdT 2

;f

nþ1

ÞXnþ1 þ ðdT

ð14aÞ

nþ1

nþ1

2

;T

nþ1

Þ

n

; c2 T ÞXnþ1 2

þ ð1  aÞDtðdT nþ1 ; c2 T;nt ÞXnþ1 2  aDtðdT nþ1 ; j2 T;xnþ1 nI ÞCI

ð14bÞ

gðnÞ ¼ Hðn  xf Þ

where we have assumed that Ch ¼ ; for simplicity, and note that the unit outward normal to X2 n ¼ nI . In the above, ð; ÞH denotes the L2 inner product over the domain H, and

which results in the approximation

BðdT; TÞ ¼ ðrdT; jrTÞX

T h ðnÞ ¼



ð1  nÞT1 þ nT2 ð1  nÞðT1 þ b1 Þ þ nðT2 þ b2 Þ

x < xf x  xf

If we interpret j and c as implicitly discontinuous across CI , the above can be expressed more compactly, which we state as the weak form: Find T 2 U0 such that

The above is clearly piecewise linear, and contains enough nþ1 nþ1 nþ1 nþ1 unknowns to match the temperature at each node as well ðdT ; cT ÞXnþ1 þ aDtBðdT ; T Þ as the point xf . Therefore, although the enrichment ¼ ðdT nþ1 ; f nþ1 ÞXnþ1 þ ðdT nþ1 ; cT n ÞXnþ1 function is itself discontinuous, the resulting approximaþ ð1  aÞDtðdT nþ1 ; cT;nt ÞXnþ1 tion T h need not be. In fact, the iterative strategy described in Sect. 3.3 weakly ensures the continuity of the approxiþ aDtðdT nþ1 ; q
nþ1 ÞCI 8 dT 2 U0 : ð15Þ mation by virtue of satisfying the constraint on the phase boundary. In addition, the above enrichment function is We note that the above equations explicitly involve the not linearly dependent with the classical shape functions. time derivative of the temperature field. With a standard

formulation, these are obtained in a straightforward fashion once the new temperature field is determined via

For the present problem, knowing all quantities at time t n , we wish to determine the temperatures T nþ1 and interface flux (4) at the new time step such that the weakform (15) is satisfied as well as the constraint (6). For notational clarity, we will drop the superscript n from all quantities, and assume the iterative procedure is invoked in moving from time step n to n þ 1. We will also use subscripts m and m þ 1, etc., to refer to iteration number. The iterative procedure adopted by the method begins by considering two sets of solution spaces. The variables of interest are the temperature T, its value on the interface w:

T nþ1  T n  ð1  aÞDtT;tn ð16Þ aDt With a semi-discrete approach, the above therefore leads to an expression for the time derivative of the temperature at each node. This is not possible with an approximation of the form (10), as the set of enriched nodes J is not fixed in time. We therefore perform an L2 projection of the form ! nþ1 n n T  T þ ð1  aÞDtT ;t wþ ¼ TjCþ ; w ¼ TjCI ð20Þ ðdT nþ1 ; T;tnþ1 ÞX  dT nþ1 ; I aDt X and the heat fluxes on the interface (4). We use the com¼0 ð17Þ pact notation: T;tnþ1 ¼

to determine the field T;tnþ1 . Importantly, this projection is only required over the set of elements with enriched nodes at time steps t n and t nþ1 . We note that similar projections are often employed in space–time formulations (Froncioni et al. 1997). It remains to apply a time-step algorithm to the equation describing the motion of the interface. To this purpose, we rewrite Eq. (3) at time step t n as:

VIn ¼


n

q L

ð18Þ

We employ the Forward Euler algorithm to obtain:

xInþ1 ¼ xnI þ Dt  VIn

ð19Þ

where xI is the interface position. We note that for higherdimensional cases, the above are typically enforced in a weak sense. See for example the formulation presented in Ghosh and Moorthy (1993). We now begin to see a very simple algorithm for solving this problem taking shape. Knowing quantities at time step t n , we can determine the position of the interface at time step t nþ1 using (18) and (19) above. We could then solve for the temperatures at the new time step, T nþ1 , using (15) and for its time derivative, using (17) if only we knew the jump in interface flux (4) at t nþ1 . The approach we will use is to adopt an iterative strategy, and determine the interface fluxes such that the interface temperature equals the melting temperature, satisfying the constraint (6).

v ¼ ðT; wþ ; w ; qþ ; q Þ;

with q
¼ q  qþ

ð21Þ

to define the solution spaces. Specifically, we let:

Ad ¼ fv satisfying ð15Þg

ð22Þ

and

I ¼ fv satisfying ð6Þg

ð23Þ AI

The goal is then to find the element v located at the intersection of Ad and I, represented geometrically in Fig. 5. We use the superscript A and I to denote an element v in Ad or I, respectively. The iterative strategy begins with an initial element vA0 in Ad and builds a sequence of approximate solutions vA1 , . . ., vAm , vAmþ1 , . . . until convergence. A given iteration vAm ! vAmþ1 involves two steps: vAm ! vIm and vIm ! vAmþ1 as shown in Fig. 5. The iterations stop when the ‘‘distance’’ between vIm and vAmþ1 (measured with an appropriate norm) is below a specified tolerance. Two successive approximations are always tied by a given search direction. The direction used when going ‘‘down’’ from the set Ad to the set I is denoted by EAI whereas the direction when going ‘‘up’’ from I to Ad is

3.3 The iterative procedure The LATIN method (Ladeve`ze 1998) is a general procedure for solving nonlinear problems in mechanics. One of its key features is the separation of unknowns into those that are global and linear from those that are local and possibly nonlinear. An iterative procedure marches between these two sets of unknowns until convergence is achieved. The method was adopted in Dolbow et al. (2001) in conjunction with the X-FEM to model crack growth with frictional contact acting on the crack faces. In contrast to Lagrange multipliers, it does not increase the size of the global system of equations, and is similar to the method of augmented Lagrangians developed by Simo and Laursen Fig. 5. The iterative procedure in the LATIN algorithm (1992).

343

denoted by EIA . The appropriate choice of the search directions in order to achieve (fast) convergence is an important aspect of the method (see Ladeve`ze 1998). The next two subsections describe the two steps: Going down from vAm to vIm in the search direction EAI . This step updates quantities on the interface. Going up from vIm to vAmþ1 in the search direction EIA . This step involves a global solve. 344

3.3.1 The local step: update on the interface The local update involves determining a new estimate vIm for the fluxes and temperatures on the interface given quantities obtained from a solution vAm 2 Ad . The new approximation is required to satisfy the constraint (i.e. vIm 2 I). Additional equations are provided by the search direction. To move from an element vAm 2 Ad to vIm 2 I, the search direction EAI is associated with a linear operator k0 . The search equations are then ðvIm  vAm Þ 2 EAI ( þ Aþ Iþ Aþ qIþ m  qm ¼ k0 ðwm  wm Þ on CI )  A I A qI m  qm ¼ k0 ðwm  wm Þ on CI

ð24aÞ ð24bÞ

nþ1 problem to be solved is then: Find ðTmþ1 ; wmþ1 Þ which satisfies

nþ1 nþ1 ðdT nþ1 ; cTmþ1 ÞXnþ1 þ aDtBðdT nþ1 ; Tmþ1 Þ nþ1 jCI ÞCI þ aDtðdT nþ1 ; k0 Tmþ1

¼ ðdT nþ1 ; f nþ1 ÞXnþ1

þ ðdT nþ1 ; c T n þ ð1  aÞ  DtT;nþ1 ÞXnþ1 t þ aDtðdT nþ1 ; ½qIm þ k0 wIm ÞCI



qAm

¼

k0 ðwIm



wAm Þ

ð25Þ

on CI

with the understanding that the relationship holds on both  Cþ I and CI , separately. This compact form will be used in the remainder of this section when convenient. The process of determining vIm given vAm then involves solving the above search equations in conjunction with the constraint on the interface  vIm 2 I ) wþ m ¼ wm ¼ Tm

ð26Þ

on CI

The resulting closed form equations for simple in this case. They are given by

ðqIm ; wIm Þ

A wAmþ1 ¼ Tmþ1 jCI

ð27aÞ

qIm ¼ qAm þ k0 ðwIm  wAm Þ

ð27bÞ

ð30Þ

After solving (29), the local fluxes on the interface at step m þ 1 are given by

qAmþ1 ¼ qIm þ k0 ðwIm  wAmþ1 Þ

ð31Þ

The search directions for the global and local steps, Eqs. (25) and (28), are conjugated i.e. expressed up to a sign by the same linear operator k0 . This ensures the convergence of the iterative strategy provided k0 is symmetric positive definite and some conditions on the material operator G described in Ladeve`ze (1998) are satisfied. In this paper, k0 is strictly positive and has the dimension of a thermal diffusivity over a velocity. We have obtained good results with the value

k0 ¼

Dt maxðji Þ L0 minðci Þ

where L0 is a characteristic length for the domain. The error indicator of convergence is computed after each global solve as the distance between the current and previous approximations:

g2 ¼

kvAmþ1  vIm k2 kvAmþ1 k2 þ kvIm k2

ð32Þ

with the norm

are very kvk2 ¼ ðq; k 1 qÞ þ ðw; k wÞ 0 0 CI CI

I wIþ m ¼ wm ¼ Tm

ð29Þ

In the above, we have made the substitution

The above can be written more compactly as

qIm

8 dT 2 U0

ð33Þ

When the indicator g is below some tolerance, the iteration process is stopped. Note that if g ¼ 0, vAmþ1 ¼ vIm is the exact solution. Finally, concerning the initialization, the solution vA0 is built by solving the global step, Eqs. (29–31), with qIm ¼ wIm ¼ 0 and setting vA0 ¼ ðTm ; wm ; qm Þ. In subsequent time steps, we have found that convergence is considerably accelerated by initializing vI0 to the values obtained at the end of the previous iteration.

3.3.2 The global step We wish to move from vIm 2 I to vAmþ1 2 Ad . This is accomplished through a global solve in conjunction with additional equations provided by the search direction EIA which is also associated with the k0 operator on the 3.4 interface. With a known element vIm , the search equations Matrix equations and summary of the algorithm are then We suppose that the temperature field, position of the A I IA A I interface, and interface fluxes at time step t n are known. In ðvmþ1  vm Þ 2 E ) qmþ1  qm the following, we denote the enriched basis functions by ¼ k0 ðwAmþ1  wIm Þ on CI ð28Þ wi ¼ /i g. The algorithm is as follows: This equation is used in conjunction with the weak form of 1. We calculate the velocity of the interface VIn using (3), the governing equations (15), by solving for qAmþ1 and and predict the position of the interface at time t nþ1 making the substitution in the surface integrals on CI . The using (19).

2. We build the global matrix equations. These result from 3. (a) We perform the initial global solve of (34) and initialize the values on the interface vI0 . substituting the approximation (10) into (29) and invoking the arbitrariness of the weight functions. The (b) We calculate the values corresponding to the local resulting system of equations reads step from (27). (c) The forcing term is updated to account for the new nþ1 nþ1 ðM þ aDtKÞ d0 ¼ F0 ð34Þ values on the interface, i.e. where I

Fm ¼ F0 þ Fm

The vector of classical and enriched degrees of freedom:

d0nþ1

¼

E ffTi gNi¼1 fbj gNj¼1 gT

Mass matrix:



M1 M¼ M3

M2 M4

where

ð35Þ

 ð36Þ

FI F ¼ 1I F2 and

Z

 ð43Þ

½FI1 i ¼ aDtð/i ; ½qIm þ k0 wIm ÞCI ½FI2 i

where, defining the bilinear form

Mðu; vÞ ¼



I

ð42Þ

¼

aDtðwinþ1 ; ½qIm

þ

k0 wIm ÞCI

ð44aÞ ð44bÞ

(d) We then solve the given system ðM þ aDtKÞ

ucv dX

nþ1 nþ1 ¼ Fm . dmþ1

X

the entries of the above are given by

½M3 ij ¼ Mðwinþ1 ; /j Þ; Stiffness matrix:



K1 K¼ K3

K2 K4



(e) The iteration error is calculated from (32). If the error is below the tolerance, we proceed to step 4. Otherwise, we return to step 3(b). ½M4 ij ¼ Mðwinþ1 ; wjnþ1 Þ 4. The temperature field and fluxes at the new time step are stored. We perform the L2 projection to determine ð37bÞ the time derivatives of the temperature field for the new time step. The matrix equations for this operation are given by

½M2 ij ¼ Mð/i ; wjnþ1 Þ

½M1 ij ¼ Mð/i ; /j Þ;

ð37aÞ

ð38Þ

Ps ¼ s

where, defining the bilinear form

Kðu; vÞ ¼

Z

u;x jv;x dX þ

X

where The vector of unknowns:

Z

k0 uv dCI

s ¼ ffai gfcj ggT

CI

½K3 ij ¼

Kðwinþ1 ; /j Þ;

½K2 ij ¼ Kð/i ; wjnþ1 Þ ½K4 ij ¼

ð39aÞ

Kðwinþ1 ; wnþ1 Þ j

Initial forcing term:

F F0 ¼ 1 F2 with

P1 P3

P2 P4



½P1 ij ¼ ð/i ; /j ÞX ; ½P3 ij ¼ ðwnþ1 ; /j ÞX ; i



½F1 i ¼ ð/i ; f nþ1 ÞXnþ1

þ ð/i ; c T n þ ð1  aÞ  DtT;nt ÞX



P¼

ð47Þ

with

ð39bÞ 

ð46Þ

The projection matrix:

the entries of the above are given by

½K1 ij ¼ Kð/i ; /j Þ;

ð45Þ

ð40Þ

ð48aÞ

½P4 ij ¼ ðwnþ1 ; /nþ1 ÞX ð48bÞ i j

The history term



ð41aÞ

½P2 ij ¼ ð/i ; wjnþ1 ÞX

s s¼ 1 s2 with



ð49Þ

  1 ð/i ; T nþ1  T n þ ð1  aÞT;nt ÞX ð50aÞ aDt ð41bÞ  1 nþ1  nþ1 ðwi ; T  T n þ ð1  aÞT;nt ÞX ½s2 i ¼ n aDt In the above, the temperature field at time step t is obtained from (10). The time derivative of the ð50bÞ temperature is expressed similarly, i.e. 5. We return to the first step and the algorithm repeats. X X T;nt ðxÞ ¼ ai /i ðxÞ þ cj /j ðxÞgðxÞ i2I j2J Remarks where the constant coefficients ai and cj are determined 1. The algorithm avoids the need to post-process the approximation to obtain the jump in the gradient of T from the projection, whose matrix form is provided at the interface by obtaining the flux q
. below in (45).

½F2 i ¼ ðwinþ1 ; f nþ1 ÞXnþ1

þ ðwinþ1 ; c T n þ ð1  aÞ  DtT;nt ÞX

½s1 i ¼

345

346

2. The use of the jump in heat flux q
to evaluate the interface velocity ensures heat conservation. This is in contrast to post-processing techniques (see Lynch and Sullivan 1985). 3. Constraints that involve the interface velocity are easily imposed through the relationship (3). 4. A large portion of the mass (36) and stiffness matrix (38) consist of the terms M1 and K1 , respectively. Importantly, these entries only need to be built at the first time step. 5. In the absence of constraints on a phase boundary, step 3 may be skipped entirely. 6. The projection is only required for elements with enriched nodes. On the rest of the domain, (16) can be used to obtain the time derivatives of the temperature in terms of the nodal coefficients Tin and Tinþ1 .

iterative strategy outlined previously. The X-FEM approximation for the temperature field is taken as

T h ðx; tÞ ¼

X

Ti ðtÞ/i ðxÞ þ

i2I

X

bj ðtÞ/j ðxÞgðx; tÞ

ð54Þ

j2J

where

gðx; tÞ ¼ e½xxfront ðtÞ

2

ð55Þ

and

J ¼ fj 2 I : xj \ xfront ðtÞ 6¼ ;g

ð56Þ

which results in no more than two nodes being enriched at any given instance. When xfront is located strictly on an element interior, for example, only the two nodes of that element are enriched. This problem obviously represents a simple case where the enrichment can be inferred by examining the form of the exact solution or forcing function. A more general case is provided in Sect. 2. In order to quantify the error in the numerical results, we use the following norms:

4 Numerical examples In this section, we consider two model one-dimensional problems for study. The first problem involves a moving Zl

2 heat source in a homogeneous domain, and serves to h 2 h kT  T k TðxÞ  T ðxÞ dx ð57aÞ 2 ¼ L illustrate the ability of the X-FEM to capture highly 0 localized solutions. As the problem has an analytic solution, it also provides a means to investigate the characZl i2

2 h teristics of the time stepping algorithm. The second h 2 TðxÞ  T h ðxÞ þ T;x ðxÞ  T;xh ðxÞ dx kT  T kH 1 ¼ example concerns the classical two-phase Stefan problem, 0 and it illustrates the accuracy and robustness of the full iterative algorithm presented in the previous section. ð57bÞ and report normalized, or relative norms.

4.1 A moving heat source problem We consider the following BVP: cT;t ¼ jT;xx þ f on x 2 ½0; l; t 2 ½0; tf  ð51aÞ Tð0; tÞ ¼ 0; Tðl; tÞ ¼ 0 ð51bÞ p  x 2 T0 ðxÞ ¼ sin ð51cÞ þ e½xxf 0  l with a solution taking the form  p  x  2 þ e½xxfront ðtÞ  et Tðx; tÞ ¼ sin ð52Þ l where xfront is used to characterize the location of a moving heat source. We assume a constant velocity Vfront for the moving source, such that its location is given by xfront ðtÞ ¼ xf 0 þ Vfront  t

ð53Þ

Given the above and (52), the forcing term f ðx; tÞ is determined such that (51) is satisfied (i.e. f ¼ cT;t  jT;xx ). The solution takes the form of a sum of an evanescent term in time (the sine) and a moving term or ‘‘spike’’. The first term is easily represented with classical piecewiselinear shape functions. The same can be said about the second term, provided the region about xfront is sufficiently refined. As this location is moving, the use of local enrichment with the X-FEM provides a much more efficient alternative. We re-emphasize that this problem does not contain a phase boundary and so it does not require the use of the

4.1.1 Numerical results The domain ½0; 500 mm is partitioned into 10 equally spaced elements. The initial location of the heat source is taken to be xf 0 ¼ 125 mm, and we set the velocity to 250 mm/s. For all of the results presented for this problem, the material parameters are set such that c=j ¼ ðp=lÞ2 . A comparison of the exact and X-FEM solutions at three different time steps is shown in Fig. 6 with a ¼ 0:55 and a time step of 0.025 s on ½0; 1:25. The X-FEM captures the moving source term extremely well, and no adverse effects were exhibited by the time projection. We did observe minor oscillations in the results (less than a 1% change in the error norm) as the location of the heat source passed across element boundaries, but the approximate solution recovered within one subsequent step. We note that the width of the ‘‘spike’’ is much smaller than the element length. Without enrichment, a classical approximation with the same mesh completely misses the contribution to the solution from the moving heat source. We repeated the simulation with a much greater source velocity of 2000 mm/s, while keeping the time step the same. The results are shown in Fig. 7. In this case, xfront is located in a different element in each time step, and there appears to be no degradation in the solution accuracy. A degradation was only noticed when the velocity exceeded h=Dt, but this is not unexpected and is easily remedied by decreasing the time step.

347

Fig. 8. Variation in error norms with a. The error norms were integrated over the time interval examined Fig. 6. Comparison of X-FEM and exact solutions at progressive time steps. The circles indicate nodal locations. Vfront ¼ 250 we have not performed extensive testing. Regardless of the mm/s

choice for a, less than a 2% error accumulated in all cases. Finally, we compare the results to those obtained using a classical finite element approximation. The results have been computed for 50 time steps on the interval ½0; 1:25, and we report the L2 and H 1 error norms integrated over time in Table 1. In the case of the classical approximation with 20 elements, the method totally failed in capturing the solution near the front location. This is not surprising as the local feature is contained within a single element. It is only with a significant refinement that the error in the classical approximation begins to approach that of the X-FEM.

4.2 Two-phase Stefan problem The classical Stefan problem models the one-dimensional freezing of a semi-infinite ðx > 0Þ domain. The boundary value problem of interest is the one-dimensional analog of (1) and (3). The initial temperature T0 ðxÞ is taken to be a constant value above the melting temperature T0 ðxÞ ¼ T2 > Tm , modeling a region that is entirely in the liquid phase. For t  0, the temperature at x ¼ 0 is set to a level T1 < Tm , causing nucleation of a phase boundary. The analytical solution describing the subsequent motion of the phase front xf is well known and given by pffiffiffiffiffiffiffi Fig. 7. Comparison of X-FEM and exact solutions at progressive xf ðtÞ ¼ 2k‘ b1 t ð58Þ time steps. V ¼ 2000 mm/s front

where b1 ¼ j1 =c1 is the thermal diffusivity of the solid phase, and the constant k satisfies the following relationship: We next examine the influence of the parameter a on the accuracy of the results. Simulations were performed for 50 pffiffiffi 2 pffiffiffi k2 j2 gðT2  Tm Þegk kL p time steps with Dt ¼ 0:025 s and Vfront ¼ 250 mm/s using e ¼ þ the same mesh of 10 elements. Figure 8 shows the variation erfðkÞ j1 ðTm  T1 ÞerfcðkpffiffigffiÞ c1 ðTm  T1 Þ in the normalized error norms integrated over the time interval as a function of a. These results illustrate that the with g ¼ b1 =b2 being the ratio of the thermal diffusivities. most accurate results are obtained when a ¼ 1:0, although The temperature field in the solid phase x  xf is then

Table 1. Comparison of error between the X-FEM and classical finite element approximations to the moving heat source problem

348

# of elements 20 100 200 400

X-FEM Rt h 0 kT  T kL2 dt

Rt

0.0075

0.0113

0

kT  T h kH 1 dt

FEM Rt h 0 kT  T kL2 dt

Rt

0.0859 0.0790 0.0488 0.0140

0.1214 0.1168 0.0999 0.0872

0

kT  T h kH 1 dt

Table 2. Numerical data of thermal properties of water saturated sand Properties

Volumetric heat capacity (cal/ C cm3 ) Thermal conductivity (cal/cm s  C Melting temperature ( C) Volumetric latent heat of fusion (cal/cm3 )

Phase

Interface

Solid

Liquid

0.49

0.62

9:6  103

6:9  103

! Tm  T1 x erf pffiffiffiffiffiffiffi T ¼ T1 þ erfðkÞ 2 b1 t

0.0 19.2

ð59Þ

and in the liquid phase x  xf

! T2  Tm x T ¼ T2  pffiffiffi erfc pffiffiffiffiffiffiffi erfcðk gÞ 2 b2 t

ð60Þ

In the present investigation, we use the properties pertaining to water-saturated, dense sand provided in Lynch and O’Neill (1981) and listed in Table 2. T1 and T2 were set to 10  C and 4:0  C. In this case, k ¼ 0:3073. We consider the case of an initial condition corresponding to t0 > 0 and the temperature field matching the exact solution given above. To approximate an infinite domain, we simulate the evolution of the temperature field on X ¼ ½0; 10 cm. We therefore expect good correlation with the exact solution at early times, and to match a steadystate solution as t ! 1. The steady-state solution is easily recovered in this case by considering the solution to (1) and (3) when the phase boundary becomes stationary. At steady state, the solution is piecewise linear with xf ¼ 7:77 cm. The domain is partitioned into 20 uniformly spaced elements of length h. In the following, we report results using a ¼ 1:0 and Dt ¼ h2 =b1 . An X-FEM approximation identical to (54) is employed, but here the enrichment function gðxÞ is taken as the Heaviside

Fig. 9. Comparison of exact and numerical solutions to the two-phase Stefan problem at times A t ¼ 180 s, B t ¼ 626 s, and C steady-state

excellent agreement with the exact solution is obtained. Figure 10 shows the front position as a function of time for both the X-FEM and exact solution from (58). It bears emphasis that the phase boundary motion does not require the direct measurement of temperature gradients at the interface. Rather, the phase boundary velocity is determined from the heat fluxes arising from the iterative procedure. Even with the relatively coarse mesh, excellent agreement is obtained until approximately 3000 s, at which point the influence of the finite geometry becomes apparent. Figure 11 plots the percent error in the front position in time. For early times, we observe less than a 2% error. Initially, the numerical phase boundary is moving slightly faster than the exact solution. We also note that the minor oscillations in the error (less than 1%) correlate with the phase boundary passing directly over a nodal position. gðxÞ ¼ Hðx  xf Þ It bears emphasis that the ability of the X-FEM to In the iterative procedure to satisfy the constraints on the capture the local solution at the phase boundary is a key phase boundary, we found that a value of 10k0 resulted in feature of the method. However, the choice of a suitable less than 15 iterations to provide a 1% error. enrichment function is clearly important. To illustrate this The results of the simulation are shown in Fig. 9 at the point, we repeat the above simulation using the enrichthree different times indicated. The enriched approxima- ment function tion is clearly capable of capturing the discontinuity in gðxÞ ¼ 1  kx  xf k temperature gradient at the phase boundary, and an

349

Fig. 10. Comparison of X-FEM front position on a finite domain Fig. 12. Comparison of X-FEM front position on a finite domain to exact solution on an infinite domain to exact solution on an infinite domain using an alternative enrichment function

Fig. 11. Percent error in the front position with time

which resembles a ‘‘tent’’ function that peaks at the phase boundary. The comparison between the X-FEM using this alternative enrichment function and the exact solution for the phase boundary location is shown in Fig. 12. The results are clearly not as accurate as those shown in Fig. 10, and the percent error approaches 10% at early times as shown in Fig. 13.

Fig. 13. Percent error in the front position with time using an alternative enrichment function

use of the X-FEM, a method that extends a classical finite element approximation with enrichment functions that suit the problem of interest. We have demonstrated that the enrichment functions can be developed to both capture a highly localized solution near a moving heat source or the discontinuity in temperature gradient at a phase boundary. The iterative strategy adopted from the LATIN method 5 allows for the enforcement of constraints on arbitrary inSummary and concluding remarks terfaces. Excellent agreement with analytical solutions was The method presented in this paper allows for accurate obtained by the method in all cases considered. solutions to transient thermal and phase change problems To our knowledge, the present formulation represents on fixed finite element meshes. This is effected through the the first attempt at representing sharp interfaces on fixed

finite element meshes. Future work will focus on the extension of the present formulation to multi-dimensional problems. We suggest that the approach taken by the X-FEM lends itself much better to multi-dimensional problems than moving-mesh algorithms as there are no concerns about element distortion. By the same token, the possible geometric complexity of the phase boundary in higher dimensions requires the use of a front capturing method. This is presently under development in conjunction with the X-FEM (Ji et al. 2002). 350

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