Remark on notation

Notation can become quite messy and confusing for the fully discretized ALE weak for-mulations. Indeed, the notation has to capture both the discretizetion in space with FEM and the discretization in time with (usually) some variation of finite difference method.

Furthermore, the evolution of the domain has to be discretized as well, consequently making differential operators operating with respect to certain configuration. Finally, any discretized expression takes place on domain at a certain time instant. For example, it has already been stated that any expression with the hat operator takes place on the reference configuration.

The domain discretization

Discretization in space of domain Ω is indicated with index h, i.e. Ω_h. Ω_h = Ω_h(t) is continuous function of time t. Time interval [0, T ] is partitioned in finite number of segments, 0 = t0 < t1 < · · · < tN = T ,

[0, T ] =

N

[

n=1

[t_n, t_n−1].

At certain time, say at t = t_n∈ [0, T ], domain and its discrete counterpart are denoted by

Ω(tn) = Ωⁿand Ωh(tn) = Ωⁿ_h.

Ω^n+1/2_h = Ω_h(t_n+1/2) denotes the discrete domain at time t_n+1/2 = ¹₂(t_n+ t_n+1). Trian-gulation of Ωⁿ_h is denoted by Tⁿ_h and triangulation of bΩ_his denoted by bT_h.

The discrete functions notation

In general, if there is an index h attached to any scalar, vector or tensor field, it denotes that it is taken from some finite element space. All finite element spaces are indicated

doi:10.6342/NTU202003676 with index h. For example,

u ∈ H¹(Ω) and uⁿ_h ∈ Vh(Tⁿ_h), Tⁿ_h = Th(Ωⁿ_h),

where V_h(Tⁿ_h) ⊂ H¹(Ωⁿ_h) is an ambient finite element space. Superscript n denotes that discrete function uⁿ_h is evaluated at time t_n. It is clear that uⁿ_h is defined on Ωⁿ_h from the context. In case the test or basis functions are dealt with, usually denoted with small Greek letters, superscript is omitted. Reason for this is that test functions are time independent (in sense of ALE time derivative, see also Section 1.1.4) and they can only appear in the context of specific configurations. For example, in expression

Z

Ωⁿ⁺¹_h

ψ_huⁿ⁺¹_h dx − Z

Ωⁿ_h

ψ_huⁿ_hdx , ψ_h ∈ V_h,

it is clear that ψ_h ∈ V_h(Tⁿ⁺¹_h ) in the first integral and ψ_h ∈ V_h(Tⁿ_h) in the second integral. Thus, unless there is a possible ambiguity, the indication of triangulation finite element space is built over is dropped and it is always clear from the context whether Vh = Vh(Tⁿ_h) or Vh = Vh(Tⁿ⁺¹_h ).

Discrete counterparts of functions defined on Q_T carry a superscript to indicate at which time they are evaluated, i.e.

uⁿ_h = u_h(t_n), and uⁿ_h is defined on Ωⁿ_h.

It is often the case that a field uⁿ_h defined on Ωⁿ_h is needed in context of Ωⁿ⁺¹_h (during the discretization step) or bΩ_h. In this case, uⁿ_h has to be composed with the ALE map at the appropriate time instance.

The discrete ALE map and related fields

Discrete ALE map, its Jacobian and gradient (and any other ALE related field), are in-dexed by two indices: h denoting that field is taken from finite element space, and n

denoting that field is evaluated at time t_n ∈ [0, T ]. For example,

Ab_h,n∈ A_h( bT_h) evaluated at t = t_n,

Jb_h,n defined on bΩ_h and evaluated at t = t_n, Fb_h,n defined on bΩ_h and evaluated at t = tn, Fb_h,n defined on bΩ_h and evaluated at t = t_n.

Return now to the function uⁿ_h which is for discretization purposes needed on Ωⁿ⁺¹_h . Then

uⁿ_h◦ bA_h,n◦ bA^-1_h,n+1 defined on Ωⁿ⁺¹_h ,

so it is introduced

A^{[n+1 ,n]}_h = bA_h,n◦ bA^-1_h,n+1

and

uⁿ_h,n+1 = uⁿ_h ◦ A^{[n+1 ,n]}_h .

In previous equations, it is also allowed (uⁿ_h)^[n+1,n] for consistency. For any field time indicator in subscript denotes the domain field is defined on, i.e. composition with appro-priate ALE maps. If there is no time indicator in subscript, it means it is equal to time indicator in superscript. The ”hat” operator over rules any compositions one might have in mind and simply means field is defined on reference domain. For example

uⁿ⁺¹_h = uⁿ⁺¹_h,n+1is defined on Ωⁿ⁺¹_h and evaluated at t = t_n+1, ubⁿ⁺¹_h is defined on bΩ_h and evaluated at t = t_n+1.

Although this notation might seem unnecessary complex and confusing at first, it equips us with an elegant and, more importantly, an unambiguous way of providing all of the

doi:10.6342/NTU202003676 necessary information on certain field using only few indices.

Semi–discrete fields

As mentioned above, subscript h denotes the space discretization. In case there is no sub-script h for certain expression, it means discretization in space hasn’t been performed. In case there is only subscript h and no subscript or superscript indicating time discretization, it means discretization is only performed in space and not performed in time variable. For example

u_h = u_h(t) ≡ field u is discretized in space but continuous in time

Ωⁿ⁺¹≡ the domain is evaluated at time t = tn+1 and not discretized in space

1.2.4 Example

In the following concrete example, conservative/non–conservative terminology is illus-trated and the strength of the conservative formulations is emphasized.

Consider the following diffusion (or heat) equation:

∂_tu − ∆u = 0 in Q_T

∇u · n = 0 on ∂Ω, t ∈ (0, T ) u(0) = u₀ in Ω₀.

(1.56)

Assume that the domain motion is a priori prescribed, i.e. the domain velocity is known and let it be defined by

w = sin 2πt







y cos πx

1

2y²π sin πx





, in Q_T. (1.57)

Note that div w = 0 in Q_T. Furthermore, let bΩ = Ω₀ where

Indeed, integrating the equation (1.56)₁over Ω, performing the integration by parts on the diffusion term, and extracting the time derivative in front of the integral sign, it is obtained

0 =

where the no–flux Neumann boundary condition have been employed.

In its ALE–non–conservative form the equation (1.56) reads:

∂

From equation (1.59) the following non–conservative, weak formulation is obtained:

find u : Q_T → R such that ∀ψ ∈ H¹(Ω) during the process of integration by parts. Extracting the temporal partial derivative in

doi:10.6342/NTU202003676 (1.60) in front of the integral sign, conservative weak formulation is obtained:

find u : Q_T → R such that ∀ψ ∈ H¹(Ω)

where Neumann boundary condition is modified in order to fit the conservative weak formulation,

[u w +∇u] · n = 0 on ∂Ω, t ∈ (0, T ).

See also Remark 1 and discussion at the end of Section 1.1.5. Term ψ div(u w) in weak formulation (1.61) can be rewritten in two different ways in order to make FEM imple-mentation feasible:

Employing the first expansion, boundary integral in (1.61) will vanish.

To make a transition from weak formulations (1.60) and (1.61) to FEM formula-tions, geometry has to be discretized and function spaces replaced with their finite–

dimensional polynomial subspaces. First, the domain Ω (circle) is replaced by its polyg-onal counterpart, Ω 7→ Ω_h. Then a triangulation on Ω_h is established, Ω_h 7→ T_h. Fi-nally, a function space V_h(T_h) ⊂ H¹(Ω_h) is chosen. For this example, the simulations are performed for the choices V_h(T_h) = P1(T_h) and V_h(T_h) = P2(T_h). The most simple time–discretization scheme is chosen for the discretization of temporal deriva-tives, namely, implicit Euler method which is first order accurate and unconditionally stable (at least for fixed mesh problems). The interval [0, T ] is uniformly partitioned, 0 = t₀ < t₁ < · · · < t_N = T , with ∆t = t_n+1− t_n.

Then, the following FEM formulations are obtained.

• FEM formulation of non–conservative weak formulation (1.60) reads: for given

uⁿ_h ∈ V_h(Tⁿ_h) find uⁿ⁺¹_h ∈ V_h(Tⁿ⁺¹_h ) such that ∀ψ_h ∈ V_h(Tⁿ⁺¹_h )

• FEM formulation of non–conservative weak formulation (1.60) employing the method of characteristics for temporal derivative, reads: for given uⁿ_h ∈ V_h(Tⁿ_h) find uⁿ⁺¹_h ∈ V_h(Tⁿ⁺¹_h ) such that ∀ψ_h ∈ V_h(Tⁿ⁺¹_h )

• FEM formulation of conservative weak formulation (1.61) employing (1.62)₁ for the ALE term expansion reads: for given uⁿ_h ∈ V_h(Tⁿ_h) find uⁿ⁺¹_h ∈ V_h(Tⁿ⁺¹_h )

• FEM formulation of conservative weak formulation (1.61) employing (1.62)₂ for the ALE term expansion reads: for given uⁿ_h ∈ V_h(Tⁿ_h) find uⁿ⁺¹_h ∈ V_h(Tⁿ⁺¹_h )

doi:10.6342/NTU202003676

The essential difference between (implicit Euler scheme) discretized conservative and non–conservative weak formulations is the domain over which integration is performed.

In FEM formulations (1.63) and (1.64) integration is performed only over Ωⁿ⁺¹_h despite the fact that uⁿ_h appears in the formulations, which is a function defined on Ωⁿ_h. On the other hand, in FEM formulations (1.65) and (1.66), three different domains appear, namely Ωⁿ_h, Ω^n+1/2_h and Ωⁿ⁺¹_h . Setting aside for the moment the integral over Ω^n+1/2_h , it can be noted that the functions are integrated over the domains they are defined on. The integration of terms involving the mesh velocity w_his performed over Ω^n+1/2_h in order not to violate the space conservation law. This issue is dealt with in detail in Chapter 3 and it is characteristic for the conservative weak formulations. Thus, at first glance, it seems that all four discretizations should produce a physically reasonable solution. Specially, the conservation property (1.58) is expected to hold on the discrete level, at least up to some extent.

Figure 1.4 shows the gain/loss of volume arising from the mesh movement. It can be seen that volume is preserved up to the order of 10⁻³. Since velocity is divergence free, clearly part of the error is coming from the artificial source term due to incorrect mesh movement. However, the volume error is relatively small and one should still expect similar solutions for the both FEM formulations. In Figure 1.5 variation of u is shown during the simulation. As proved above, analytical variation is zero at all times. However, Figure 1.5 clearly illustrates the superiority of conservative formulations in this regard.

While conservative formulations produce solution uh with the variation up to order of

Figure 1.4: The domain volume gain and loss during the simulation due to mesh move-ment.

(a) Variation of uh employing the conservative FEM formulation (1.65).

(b) Variation of uh employing the conservative FEM formulation (1.66).

(c) Variation of uh employing the conservative FEM formulation (1.63).

(d) Variation of uh employing the conservative FEM formulation (1.64).

Figure 1.5: Variation of u over time for various FEM formulations. Finite element space is chosen as V = P1. f (t) =R

Ωu dx denotes the variation of u over Ω at time t.

10⁻¹⁴, non–conservative formulations produce solution uh with the variation up to the order of only 10⁻³.