The Finite Element Method

The formulation of the equilibrium equations governing the response of a continu-ous system under specific loads and constraints at its boundaries is usually provided in the form of a differential equation, e.g. Eq.1.12. The differential equation is also known as the strong form of the problem. The strong form requires strong conti-nuity on the dependent field variables (velocity and pressure, in our case) such that the functions defining these variables have to be differentiable up to the order of the partial differential equations that are in the strong form of the system equations.

Instead of trying to obtain the exact solution for a strong form of the system equa-tions, we can use the finite element method on a weak (or variational) form. It is worth pointing out that a solution to the strong form will also satisfy the weak form, but not vice versa. Since the weak form uses a lower order of derivatives, it can be satisfied by a larger set of functions. One of the most common approaches used to transform strong into weak form of the system is the Galerkin method where the best approximate solution is found within a family of trial functions through weighting the governing equations with test functions.

For example, when applied to the glacier flow problem, we need to identify the field variables as well as to obtain the strong form of the governing continuum

me-chanics equations and boundary conditions. Substitute τ in Eq.1.12 with Eq.1.3 and introduce a strain rate dependent viscosity ν( ˙) = A⁻ⁿ¹ ˙

1 n−1

e , we arrive at the following expression

−∇ ·

ν( ˙) ˙

+ ∇p = ρg (1.15)

Since strain rate is also a function of the velocity, the Stokes equation can be written in terms of velocity:

−∇ ·

ν(u) ˙(u)

+ ∇p = ρg (1.16)

where two variables, velocity u and pressure p, are to be solved. Now that the field variables are identified, we can establish the weak form of Eq.1.13 and 1.14 using the Galerkin procedure. The momentum equation is weighted with velocity test functions v and the continuity equation is weighted with pressure test functions q. Integrating over the domain of interest Ω (with ∂Ω representing boundaries of the domain) and using the divergence theorem to lower the order of the derivatives in the expressions,

−

where n is the normal vector pointing out of the boundaries and σ · n is the traction on the boundaries and can be conveniently defined. The same procedure can be

performed on the incompressibility equation,

Z

Ω

q(∇ · u)dx = 0 (1.22)

Now that we’ve obtained the weak forms of the governing equations (Eq.1.21 and 1.22), a set of boundary conditions are needed in order to complete the definition of the problem.

Before moving into the next step, there are a few simplifications that need to be made:

1. Real world glaciers are three-dimensional. However, it would be computation-ally expensive to build a 3D Finite Element model that matches the size of a glacier.

Therefore, we choose to build a two-dimensional model along a central flow line and solve the equations within a vertical cross-section that cuts through the glacier, where the direction along the flow is x and the vertical direction is z. Moreover, since calv-ing events only happen at the terminus of a glacier, we focus on the portion of the glacier near the terminus and make the initial length to thickness ratio at 6, which will be justified later.

2. Glaciers can usually be divided into three different layers: snow, firn, and ice, from top to bottom and in the order of increasing density. Once fallen onto the surface of the glacier, snow starts to recrystallize and slowly turns into firn. After all the air and water pockets are sealed off, firn becomes glacier ice. Moreover, each glacier has its own density profile, which depends on factors like temperature, size, and annual precipitation. Since our model is generalized rather than focused on one specific location, the density of ice used in our model is taken as 910 kg/m³, appropriate for glaciers thinner than 1000 m (Cuffey and Paterson, 2010).

3. We are also treating the modeled glacier as an isothermal object, but the temperature profile of real world glaciers has a depth dependence (e.g., Blatter , 1987;

Price et al., 2002). The snow/ice temperature near the surface is largely dependent on the surface air temperature, which is colder in winter and colder in polar or high altitude regions. On the other hand, the ice temperature at the bottom of the glacier is controlled by the bed, where both geothermal heat flux and frictional heating can occur, and is typically warmer than the part of glacier closer to the surface. The temperature within a glacier ranges from 0^◦C to −40^◦C (Cuffey and Paterson, 2010) and we have taken an averaged value of −20^◦C (253.39 K). This is a conservative approximation because tidewater glaciers may be warmer and thus less viscous (Eq.1.10 indicates that viscosity decreases with increasing temperature).

There are two types of boundary conditions that are commonly used in a con-tinuum mechanic problem: Neumann (or natural) boundary conditions and Dirichlet (or essential, geometric) boundary conditions. The former are imposed on the sec-ondary variables like forces and tractions while the latter are imposed on the primary variables like displacements. The four boundaries of our two-dimensional model are:

upstream, downstream, surface, and bed. The upstream boundary will have a flux of ice flowing into the interested area and can be defined as a Dirichlet boundary condition that regulates the velocity. The downstream boundary is where the glacier becomes in contact with the ocean and will have a Neumann boundary condition im-posing a traction balanced with hydrostatic pressure for the part that’s underwater.

The surface of the glacier is exposed in air and will be under atmospheric pressure, which is usually an order of magnitude smaller than that within a glacier. Hence, a Neumann boundary condition can also be applied here which denotes a traction-free surface. The bottom of the glacier is in contact with bed rock and can be anywhere between completely frozen (no slip) to well-lubricated (free slip). The no slip and free slip boundary conditions is a Dirichlet boundary condition and comes in the form of velocity regulation, with both components of velocity vanishing for no slip and only the normal component vanishing for free slip. The four boundary conditions are listed

below:

u = (constant₁, constant₂) upstream, ∂Ω_u (1.23) σ · n = (water pressure)n downstream, ∂Ω_d (1.24)

σ · n = 0 surface, ∂Ω_s (1.25)

u = 0 or u · n = 0 bed, no slip or free slip, ∂Ω_b (1.26)

The Dirichlet boundary conditions can be directly defined but the Neumann boundary conditions need to be incorporated into the weak form of the governing equations and Eq.1.21 and 1.22 can be combined into one equation

Z

where u and v are trial and test functions of velocity, p and q are trial and test functions of pressure, C is a user defined constant to enforce incompressibility, ρ_i and ρ_w are the density of ice and water respectively, g is the magnitude of gravitational acceleration, g is the gravitational acceleration vector and points downwards (−z direction), D is the vertical position of the waterline, and z is the vertical position of the nodes on the downstream boundary.

The solution to the above equation can be derived through discretization. The domain of interest Ω is divided into N smaller elements (Ω1, ..., ΩM), with a total of M nodes. An integral on the entire domain Ω now becomes a sum of integrals within each element Ω_i. For example

Z

Similarly, the solution can be approximated by a linear combination of a group of basis functions defined on the nodes.

u ≈ u^h =

M

X

j=1

U_jφ_j (1.29)

where U_j are shape functions and φ_j are nodal basis functions. For a single variable linear equation, the weak form can be easily written in a matrix form and apply solvers on. However, there are two variables in Eq.1.27, what we call a mixed form.

It is more challenging in the sense that choosing the appropriate test function spaces as well as enforcing piecewise conservation of mass are hard to be achieved at the same time. Moreover, the fact that the mixed variational form leads to indefinite matrix equations would require pressure-stabilizing terms to be added to the weak form (Logg et al., 2012). Therefore, we implement the open source Finite Element solver FEniCS (Alnæs et al., 2015) in our model.