Automated shape differentiation in the Unified Form Language

Physical models often involve functionals that assign real values to the solutions of partial differential equations (PDEs). For instance, the compliance of a structure is a function of the solution to the elasticity equations, and the drag of a rigid obstacle immersed in a fluid is a function of the solution to the Navier-Stokes equations.

This type of functional depends on the PDE parameters, and it is often possible to compute the derivative of a functional with respect to a chosen set of parameters. This derivative can in turn be used to perform sensitivity analysis and to run optimization algorithms with respect to parameters in the PDE.

The shape of the physical domain that is part the PDE-model (like the shape of the rigid obstacle mentioned above) is a parameter that is not straightforward to handle. Although we can compute the shape derivative of a functional following shape calculus rules (Delfour and Zolésio 2011), this differentiation exercise is often tedious and error prone. In Schmidt (2018), Schmidt introduces the open-source library FEMorph: an automatic shape differentiation toolbox for the Unified Form Language (UFL, Alnæs et al. 2014). The library FEMorph is based on refactoring UFL expressions and applying shape calculus differentiation rules recursively. It can compute first- and second-order shape derivatives (both in so-called weak and strong form), and it has been successfully employed to solve shape optimization problems (Schmidt et al. 2018).

This article presents an alternative approach to automated shape differentiation. The key idea is to rely solely on pullbacks and standard Gâteaux derivatives. This approach is more generic and robust, because it does not require handling of special cases. In particular, it circumvents the implementation of shape calculus rules, and required only a minor modification of UFL, because UFL supports Gâteaux derivatives with respect to functions. As a result, UFL is now capable of shape differentiating any integral that can be expressed in it.

This article is organized as follows. In Section 2, we revisit shape calculus and describe how to shape differentiate using standard finite element software. In Section 3, we consider three test cases and show how to compute shape derivatives using Firedrake and UFL. In Section 4, we describe code validation of this new UFL feature. In Section 5, we solve a PDE-constrained shape optimization test case with a descent algorithm implemented in Firedrake and UFL. Finally, in Section 6, we summarize the contribution of this article.

A shape functional is a map \(\mathrm {J}:{\mathcal {U}}\to \mathbb {R}\) defined on the collection of domains \({\mathcal {U}}\) in \(\mathbb {R}^{d}\), where \(d\in \mathbb {N}\) denotes the space dimension.

We follow the perturbation of identity approach (Simon 1980) and for a vector field \({\mathbf {V}}\in W^{1,\infty }({\mathbb {R}^{d}},\mathbb {R}^{d})\), we consider the family of transformations {P_s}_s≥ 0 defined by P_s(x) = x + sV(x) for every \({\mathbf {x}}\in \mathbb {R}^{d}\). Note that P_s is a diffeomorphism for any sufficiently small \(s\in \mathbb {R}^{+}\).

For a domain \({\Omega }\in {\mathcal {U}}\), let Ω_s : = P_s(Ω) and assume that \({\Omega }_{s}\in {\mathcal {U}}\) for s sufficiently small. The shape directional derivative of J at Ω in direction V is the derivative

We say that J is shape differentiable in Ω if the directional derivative dJ(Ω)[V] exists for every direction \({\mathbf {V}}\in W^{1,\infty }({\mathbb {R}^{d}},\mathbb {R}^{d})\), and if the associated map \(\mathrm {d\mathrm {J}}({\Omega }): W^{1,\infty }({\mathbb {R}^{d}},\mathbb {R}^{d})\mapsto \mathbb {R}\) is linear and continuous. In this case, the linear operator dJ(Ω) is called the shape derivative of J in Ω.

To illustrate the shape differentiation of a shape functional, we consider the prototypical example where \(u_{{\mathbf {P}}_{s}}\) is a scalar function.¹ The subscript P_s highlights the possible dependence of \(u_{{\mathbf {P}}_{s}}\) on the domain Ω_s.

The standard procedure to compute dJ is to employ transformation techniques and rewrite where DP_s denotes the Jacobian matrix of P_s and \(u_{{\mathbf {P}}_{s}}\circ {\mathbf {P}}_{s}\) denotes the composition of \(u_{{\mathbf {P}}_{s}}\) with P_s, that is, \((u_{{\mathbf {P}}_{s}}\circ {\mathbf {P}}_{s})({\mathbf {x}}) =u_{{\mathbf {P}}_{s}}({\mathbf {P}}_{s}({\mathbf {x}}))\) for every x ∈Ω. Note that \(\det ({\mathbf {D}}{\mathbf {P}}_{s})>0\) for s sufficiently small. Then, by linearity of the integral, the shape derivative dJ is given by where d˙s(⋅ ) denotes the derivative with respect to s at s = 0. The term \(\text{d}_{\text{s}}(u_{{\mathbf {P}}_{s}}\circ {\mathbf {P}}_{s})\) is often called the material derivative (Berggren 2010). Its explicit formula depends on whether the function \(u_{{\mathbf {P}}_{s}}\) does or does not dependent on Ω_s (see Section 3).

Next, we repeat the derivation of (3) in the context of finite elements and derive an alternative formula for dJ. Let \(\{K_{i}\}_{i\in \mathcal {I}}\) be a partition of Ω such that \(\dot \cup _{i} \overline K_{i} = \overline {\Omega }\) and such that the elements K_i are non-overlapping. Additionally, let \(\{{\mathbf {F}}_i\}_{i\in \mathcal {I}}\) be a family of diffeomorphisms such that \({\mathbf {F}}_i(\hat {K}) = K_{i}\) for every \(i\in \mathcal {I}\), where \(\hat {K}\) denotes a reference element. This induces a partition \(\{{\mathbf {P}}_{s}(K_{i})\}_{i\in \mathcal {I}}\) of Ω_s. To evaluate (1), standard finite element software rewrites it as Let \({\mathbf {F}}_{i}^{-1}\) denote the inverse of F_i, that is, \({\mathbf {F}}_{i}^{-1}({\mathbf {F}}_{i}({\mathbf {x}}))= {\mathbf {x}}\) for every \({\mathbf {x}}\in \hat {K}\), and \({\mathbf {F}}_{i}({\mathbf {F}}_{i}^{-1}({\mathbf {x}}))= {\mathbf {x}}\) for every x ∈ K_i. Since \({\mathbf {P}}_{s} = ({\mathbf {P}}_{s}\circ {\mathbf {F}}_{i})\circ {\mathbf {F}}_{i}^{-1}\) and P_s ∘F_i = F_i + sV ∘F_i, we can rewrite (4) as follows: Let \(\{g_{i}\}_{i\in \mathcal {I}}\) be the collection of maps defined by Then, formula (5) can be rewritten as and taking the derivative of (5) with respect to s implies Equation (6) gives an alternative and equivalent expression for the shape derivative (3). However, to derive formula (3), it is necessary to follow shape calculus rules by hand, which is often a tedious and error prone exercise. Equation (6), by contrast, can be derived automatically with finite element software. Indeed, to evaluate J(Ω), standard finite element software rewrites it as In UFL, the maps \(\{g_{i}\}_{i\in \mathcal {I}}\) are constructed symbolically and in an automated fashion. Therefore, it is possible to evaluate dJ(Ω)[V] by performing the steps necessary for the assembly of J(Ω) and, at the appropriate time, differentiating the maps \(\{g_{i}\}_{i\in \mathcal {I}}\). To be precise, this differentiation corresponds to a standard Gâteaux directional derivative, because the integrand in (6) corresponds to the following limit which can be interpreted as the Gâteaux directional derivative of g_i at T = F_i in the direction V ∘F_i (Hinze et al. 2009, Def. 1.29). This viewpoint is important to correctly implement this differentiation step in the existing pipeline in UFL (see Fig. 1). We emphasize that this also enables computing higher order shape derivatives by simply taking higher order Gâteaux derivatives in (6).

In this section, we consider three examples based on (1) that cover most applications. For these examples, we give explicit expressions of dJ using (3) and (6) and show how to compute dJ with the finite element software Firedrake² (Rathgeber et al. 2016). To shorten the notation, we define V_i : = V ∘F_i.

We validate our implementation by testing that the Taylor expansions truncated to first and second order satisfy the asymptotic conditions where and In Fig. 2, we plot the values of δ₁ and δ₂ for s = 2^− 1,2^− 2, \(\dots , 2^{-10}\), and J as in examples 1 and 3 from the previous section (we denote these functionals J₁ and J₂ respectively). The vector field V is chosen randomly. This experiment clearly displays the asymptotic rates predicted by (17).

We have repeated this numerical experiment for many other test cases, including functionals that are not linear in u, functionals given by integrals over ∂Ω involving the normal n, and functionals that are constrained to linear and nonlinear boundary value problems with nonconstant right-hand sides and nonconstant Neumann and Dirichlet boundary conditions. In every instance, we have observed the asymptotic rates predicted by (17). The code for these numerical experiments is available at “Software used in ‘Automated shape differentiation in the Unified Form Language’ ()”.

In this section, we show how to use Firedrake and the new UFL capability to code a PDE-constrained shape optimization algorithm. As test case, we consider the optimization of a pipe to minimize the dissipation of kinetic energy of the fluid into heat. This example is taken from Schmidt (2010, Sect. 6.2.3). To simplify the exposition, we use a very simple optimization strategy. At the end of the section, we will comment on possible improvements.

The initial design of the pipe is shown in Fig. 3 (top). The pipe contains viscous fluid (with viscosity ν), which flows in from the left and is modeled using the incompressible Navier-Stokes equations. To be precise, let Ω be the shape of the pipe, Γ ⊂ ∂Ω be the outflow boundary of the pipe (that is, the end of the pipe on the right), and u and p be the velocity and the pressure of the fluid, respectively. Then, u and p satisfy Here, g is given by a Poiseuille flow at the inlet and is zero on the walls of the pipe

The goal is to modify the central region of the pipe so that the shape functional is minimized. To solve this shape optimization problem, we parametrize the initial design with a polygonal mesh and update the node coordinates using a descent direction optimization algorithm with a fixed step size. As descent directions, we use Riesz representatives of the shape gradient with respect to the inner product induced by the Laplacian, i.e., at each step the deformation is given by the solution to This approach is also known as Laplace smoothing. To avoid degenerate results, we penalize changes of the pipe volume. The whole algorithm, comprising of state and adjoint equations and shape derivatives, is contained in Listing 4 and described in detail in the following paragraph. The optimized shape is displayed in Fig. 3 (bottom), the convergence history is in Fig. 4. These results are compatible with those in Schmidt (2010, Sect. 6.2.3) and clearly indicate the success of the shape optimization algorithm.

                      
                        dL = -inner(nu*grad(u)*grad(W), grad(v))*dx
                      
                      
                           -inner(nu*grad(u), grad(v)*grad(W))*dx
                      
                      
                           -inner(v, grad(u)*grad(W)*u)*dx
                      
                      
                           +tr(grad(v)*grad(W))*p*dx
                      
                      
                           -tr(grad(u)*grad(W))*q*dx
                      
                      
                           +div(W)*inner(nu*grad(u), grad(v))*dx
                      
                      
                           -div(W)*inner(div(v), p)*dx
                      
                      
                           +div(W)*inner(div(u), q)*dx
                      
                      
                           +div(W)*inner(v, grad(u)*u)*dx
                      
                      
                           +nu*inner(grad(u), grad(u))*div(W)*dx
                      
                      
                           -2*nu*inner(grad(u)*grad(W), grad(u))
                      
                      
                           *dx
                      
                    

We have presented a new and equivalent formulation of shape derivatives in the context of finite elements as Gâteaux derivatives on the reference element. While the formulation applies to finite elements in general, we have implemented this new approach in UFL due to its extensive support for symbolic calculations. This new UFL capability allows computing shape derivatives of functionals that are defined as volume or boundary integrals, and that are constrained to linear and nonlinear PDEs. During shape differentiation, our code treats finite element functions and global functions differently. This behavior is correct and necessary to handle PDE-constraints properly. In combination with a finite element software package, such as FEniCS or Firedrake, that takes as input UFL, this enables the entirely automated shape differentiation of functionals subject to boundary value problems. This notably simplifies tackling PDE-constrained shape optimization problems.

Compared to the existing shape differentiation toolbox FEMorph, our code does not compute shape derivatives in strong form because it neither relies on shape calculus differentiation rules nor performs integration by parts. However, in practice, we do not consider this a limitation as it has been shown in Hiptmair et al. (2015), Berggren (2010), and Zhu (2018) that the weak form is superior when the state and the adjoint equations are discretized by finite elements.

The code for the numerical experiments is available at Software used in ‘Automated shape differentiation in the Unified Form Language’ (2019).