Hermite elements are exact! Part 3: not quite the proof, yet!

In the previous instalment of this series, I showed that in a finite element analysis, it is crucial for the loads to be applied to the structure in a consistent way. I introduced equivalent nodal loads and claimed that, if properly evaluated, these loads produce exact nodal displacements. In the present post, we set the stage for the actual proof of this claim. The proof itself will be presented in the next post.

For an arbitrary structure, the proof is quite technical because we must handle longitudinal and transverse displacements and their continuity at each node, as well as bending, tension/compression and possibly torsion energies. In the remainder of this post, we will therefore focus on a restricted case. From this particular case to the most general case, the proof remains essentially unchanged (you have my word!), but gets considerably more intricate…

A simplified, variational setting

We consider a structure made of an assembly of beam elements that are connected at the nodes of the structure. An essential assumption is that the elastic properties of the beam elements (axial stiffness \(EA\), bending stiffness \(EI\)) are constant between each node. The problem of elastic equilibrium of the structure is analyzed with Hermite elements in a Galerkin setting. We will prove that the nodal displacements are exact under the following assumptions.

Assumption 1. The structure is rectilinear. In other words, the structure is essentially some sort of continuous beam. The total length of the structure is \(L\), and the abscissa \(0 \leq x \leq L\) is measured from its left end. The nodes of the structure are located at \(x_1 = 0 < x_2 < \cdots < x_N = L\) (\(N\): total number of nodes).

Assumption 2. The structure is subjected to transverse loads only, all located in the same plane. The distributed load has density \(p(x)\), which might be a generalized function (to the effect that transverse point loads are allowed).

As a result of the geometry and the loading, the structure will undergo transverse displacements (deflections) only. Let \(v(x)\) be the deflection of point \(x\) of the structure. In order to be admissible, the mapping \(x \mapsto v(x)\) must have \(\mathrm{C}^1\) continuity and meet all essential boundary conditions. These requirements define the space \(V\) of admissible deflections. In particular, the true deflection \(v\) belongs to this space: \(v \in V\).

Assumption 3. The space \(V\) of amissible deflections has the structure of a vector space. In general, the space \(V\) has an affine structure; the present assumption will hold if non-zero displacements (inhomogeneous Dirichlet boundary conditions) are excluded.

In order to define the Galerkin approximation to the problem of the equilibrium of the structure \(\mathcal{S}\), we need to introduce a variational setting. At equilibrium, the structure wants to minimize its total potential energy \(Π\). Under the above assumptions, we have \[ \Pi[v] = \underbrace{\int_0^L \tfrac{1}{2} \, EI(x) \, \bigl[ v''(x) \bigr]^2 \, \mathrm{d} x}_{\text{strain energy}} - \underbrace{\int_0^L p(x) \, v(x) \, \mathrm{d} x}_{\substack{\text{potential of} \\ \text{external forces}}}. \] where \(x \mapsto EI(x)\) is the (local) bending stiffness of the structure. Since \(EI\) is constant between two adjacent nodes, we write \[ \Pi[v] = \sum_{k = 1}^{N - 1} \int_{x_k}^{x_{k+1}} \tfrac{1}{2} \, EI_k \, \bigl[ v''(x) \bigr]^2 \, \mathrm{d} x - \int_0^L p(x) \, v(x) \, \mathrm{d} x, \] where \(EI_k\) denotes the bending stiffness of beam element number \(k\), located between nodes \(x_k\) and \(x_{k+1}\). The true deflection \(v \in V\) of the structure is such that \[ \Pi[v] \leq \Pi[w] \quad \text{for all} \quad w \in V. \]

It is a classical application of the calculus of variations to show that, under mild assumptions, minimizing the total potential energy \(\Pi\) is equivalent to solving the following problem \[ (\mathcal{P}) \qquad \text{Find } v \in V \quad \text{such that} \quad a(v, w) = \int_0^L p(x) \, w(x) \, \mathrm{d} x \quad \text{for all} \quad w \in V \] where \(a\) is defined as follows \[ a(v, w) = \sum_{k=1}^{N-1} \int_{x_k}^{x_{k+1}} EI_k \, v''(x) \, w''(x) \, \mathrm{d} x. \]

The precise expression of \(a\) does not really matter; what really matters is that \(a\) is a bilinear, symmetric form. Problem \((\mathcal{P})\) defines a variational problem ; in the next section, we define a Galerkin approximation to the solution of this problem.

The subspace of Hermite deflections

The idea behind Galerkin discretization is very simple: the initial variational problem \((\mathcal{P})\) requires the full exploration of the (infinite-dimensional) space \(V\). This is in general not possible! Rather, we will consider a finite-dimensional subspace \(V_{\mathrm{H}}\) and solve the following variational problem \[ (\mathcal{P}_{\mathrm{H}}) \qquad \text{Find } v_{\mathrm{H}} \in V_{\mathrm{H}} \quad \text{such that} \quad a(v_{\mathrm{H}}, w_{\mathrm{H}}) = \int_0^L p(x) \, w_{\mathrm{H}}(x) \, \mathrm{d} x \quad \text{for all} \quad w_{\mathrm{H}} \in V_{\mathrm{H}}. \]

The solution \(v_{\mathrm{H}}\) to problem \((\mathcal{P}_{\mathrm{H}})\) defines the Galerkin approximation to the solution \(w\) of the initial problem \((\mathcal{P})\). Now that we know how we are to derive an approximate solution to our structural problem, we need to define the subspace \(V_{\mathrm{H}}\). Of course, we will choose Hermite shape functions (hence the “H” subscript!).

More precisely, \(V_{\mathrm{H}}\) is the space of functions \(w_{\mathrm{H}}\colon (0, L) \longrightarrow \mathbb{R}\) such that

• \(w_{\mathrm{H}} \in V_{\mathrm{H}}\): in other words, \(w_{\mathrm{H}}\) has \(\mathrm{C}^1\) continuity and satisfies all essential boundary conditions,
• \(w_{\mathrm{H}}\) is a cubic polynomial between two successive nodes.

Hermite shape functions were introduced in the first part of this series, where it was shown in particular that the deflection of a single beam element was fully defined by the deflection and rotation of its end nodes. The same holds for a full structure: any \(w_{\mathrm{H}} \in V_{\mathrm{H}}\) is fully defined by the nodal deflections \(w_{\mathrm{H}}(x_k)\) and nodal rotations \(w_{\mathrm{H}}'(x_k)\). Therefore, \(V_{\mathrm{H}}\) is at most \(2N\)-dimensional subspace of \(V\). I wrote “at most” because boundary conditions will certainly reduce the dimensionality of \(V_{\mathrm{H}}\). Any \(w_{\mathrm{H}} \in V_{\mathrm{H}}\) can therefore be decomposed uniquely as the following linear combination \[ w_{\mathrm{H}}(x) = \sum_k w_{\mathrm{H}}(x_k) \, \Phi_k(x) + \sum_k w_{\mathrm{H}}'(x_k) \, \Psi_k(x), \tag{1}\] where the sums run over all unconstrained nodes, and \(\Phi_k\), \(\Psi_k\) are the global shape functions.

What are these equivalent nodal loads anyway?

We are now in a position to state a mathematical definition of the equivalent nodal loads. The equivalent nodal forces \(F_k\) and torques \(T_k\) are defined as follows \[ F_k = \int_0^L p(x) \, \Phi_k(x) \, \mathrm{d} x \quad \text{and} \quad T_k = \int_0^L p(x) \, \Psi_k(x) \, \mathrm{d} x. \]

It results from the decomposition (1) of any \(w_{\mathrm{H}} \in V_{\mathrm{H}}\) that \[ \int_0^L p(x) \, w_{\mathrm{H}}(x) \, \mathrm{d} x = \sum_k F_k \, w_{\mathrm{H}}(x_k) + \sum_k T_k \, w_{\mathrm{H}}'(x_k). \]

In other words, for Hermite displacements, the equivalent nodal loads produce the same work as the true loads and the discretized problem \((\mathcal{P}_{\mathrm{H}})\) reads \[ (\mathcal{P}_{\mathrm{H}}) \qquad \text{Find } v_{\mathrm{H}} \in V_{\mathrm{H}} \quad \text{such that} \quad a(v_{\mathrm{H}}, w_{\mathrm{H}}) = \sum_k F_k \, w_{\mathrm{H}}(x_k) + \sum_k T_k \, w_{\mathrm{H}}'(x_k) \quad \text{for all} \quad w_{\mathrm{H}} \in V_{\mathrm{H}}. \]

Hermite elements are exact – The trivial case

Let us assume for a moment that the true loads are applied to the nodes only. Then, between two successive nodes, the bending equation reads \[ x_k \leq x \leq x_{k+1}: \quad EI_k \, \frac{d^4 v}{dx^4} = 0, \] where \(v\) denotes the true deflection of the structure. Upon integration, it is found that \(v\) is a cubic polynomial over \((x_k, x_{k+1})\). Since \(v\) is an admissible deflection, it belongs to \(V_{\mathrm{H}}\).

In other words, when loads are concentrated at the nodes of the structure, the true solution happens to belong to the subspace \(V_{\mathrm{H}}\). Therefore, the discretized problem \((\mathcal{P}_{\mathrm{H}})\) delivers the exact solution to the initial problem \((\mathcal{P})\).

An interesting consequence is the following result: if \(v_{\mathrm{H}} \in V_{\mathrm{H}}\) is such that \[ a(v_{\mathrm{H}}, w_{\mathrm{H}}) = \sum_k F_k \, w_{\mathrm{H}}(x_k) + \sum_k T_k \, w_{\mathrm{H}}'(x_k) \quad \text{for all} \quad w_{\mathrm{H}} \in V_{\mathrm{H}}, \] then it is true that \[ a(v_{\mathrm{H}}, w) = \sum_k F_k \, w(x_k) + \sum_k T_k \, w'(x_k) \quad \text{for all} \quad w \in V. \]

The only difference between these two assertions is the space to which the test function belongs (\(V_{\mathrm{H}}\) vs. \(V\)). This will in fact make a huge difference in the next instalment of this series.