A Spline-Based Stress Function Approach for the Principle of Minimum Complementary Energy

TL;DR

This work addresses the challenge of achieving accurate stress predictions with few degrees of freedom in plane elastostatics by introducing a spline-based Airy stress function within the principle of minimum complementary energy. The method uses a B-spline representation of the Airy function on a geometric mapping to complex geometries, enabling direct enforcement of equilibrium and traction boundary conditions, with remaining degrees of freedom found via minimization of the total complementary energy $\Pi^{*}$. Validation against analytical solutions and application to a bar under self-weight, a uniformly loaded bending beam, a bi-layer anisotropic cantilever, and a non-prismatic parabolic cantilever demonstrate high fidelity stress fields with DOF reductions of several orders of magnitude compared to high-fidelity references. The approach offers a flexible, efficient tool for structural stress analysis and has potential extensions to three-dimensional problems and data-driven design or material identification contexts.

Abstract

In computational engineering, ensuring the integrity and safety of structures in fields such as aerospace and civil engineering relies on accurate stress prediction. However, analytical methods are limited to simple test cases, and displacement-based finite element methods (FEMs), while commonly used, require a large number of unknowns to achieve high accuracy; stress-based numerical methods have so far failed to provide a simple and effective alternative. This work aims to develop a novel numerical approach that overcomes these limitations by enabling accurate stress prediction with improved flexibility for complex geometries and boundary conditions and fewer degrees of freedom (DOFs). The proposed method is based on a spline-based stress function formulation for the principle of minimum complementary energy, which we apply to plane, linear elastostatics. The method is first validated against analytical solutions and then tested on two test cases challenging for current state-of-the-art numerical schemes, a bi-layer cantilever with anisotropic material behavior, and a cantilever with a non-prismatic, parabolic-shaped beam geometry. Results demonstrate that our approach, unlike analytical methods, can be easily applied to general geometries and boundary conditions, and achieves stress accuracy comparable to that reported in the literature for displacement-based FEMs, while requiring significantly fewer DOFs. This novel spline-based stress function approach thus provides an efficient and flexible tool for accurate stress prediction, with promising applications in structural analysis and numerical design.

Figures (15)

• Figure 1: Geometric mappings from the parametric domain $\hat{\Omega}$. Top: mapping to a quadrilateral domain $\Omega$ with curved boundary using a single mapping $\boldsymbol{T}$. Bottom: Mapping to a plate with a circular hole, constructed from eight patches $\Omega_1$ to $\Omega_8$ with individual mappings $\boldsymbol{T}_1$ to $\boldsymbol{T}_8$. Interfaces are indicated by dashed lines and coupled through traction equilibrium.
• Figure 2: Incorporation of traction boundary conditions for the special case $\boldsymbol{T}=\rm id$: prescribed stress components (left) and control variables for the spline-based stress function $\hat{\varphi}$ and its derivatives (right). The prescribed stress components determine the values of the marked control variables (filled circles) of the second-order derivative splines. These conditions are propagated to $\hat{\varphi}$, imposing constraints on a subset of the control variables $\hat{\phi}_{i,j}$. These constraints are indicated by lines between involved control variables. Note that, in this example, only the middle control variable of $\hat{\varphi}$ remains unconstrained.
• Figure 3: Bar under self-weight: setup.
• Figure 4: Bar under self-weight: stress distributions for the three stress components.
• Figure 5: Bar under self-weight: comparison of the stress component $\sigma_{yy}$ across the beam’s height at mid-width ($x = 0.25~m$). The gray shading, equal to one cross-section dimension ($c=0.5~m$), marks the region near the clamped top where boundary effects are expected.