Structural Mechanics#

Structural mechanics analyses are used to predict the response of structures subjected to mechanical loads and provide the physical quantities required by many optimization criteria such as compliance, stress, eigenfrequencies, and buckling loads.

Throughout this section, let \(\Omega \subset \mathbb{R}^3\) denote the computational domain and

\[\partial \Omega = \Gamma_D \cup \Gamma_N \cup \Gamma,\]

where:

  • \(\Gamma_D\) denotes Dirichlet boundaries.

  • \(\Gamma_N\) denotes Neumann boundaries.

  • \(\Gamma\) denotes traction-free boundaries.

Overview#

OpenPisco provides interfaces to several structural solvers and supports multiple mechanical analyses commonly encountered in topology optimization, shape optimization, and structural design.

Supported Analyses#

  • Linear Elasticity

  • Static Buckling

  • Modal Analysis

  • Harmonic Analysis

Linear Elasticity#

The linear elasticity analysis predicts the displacement, strain, and stress fields generated by external loads acting on an elastic structure.

Governing Equations#

The displacement field \(u\) satisfies

\[\begin{split}\left\{ \begin{aligned} \operatorname{div}(\sigma(u)) + f &= 0 && \text{in } \Omega,\\ \sigma(u)\cdot n &= g && \text{on } \Gamma_N,\\ \sigma(u)\cdot n &= 0 && \text{on } \Gamma,\\ u &= 0 && \text{on } \Gamma_D. \end{aligned} \right.\end{split}\]

where

  • \(u\) is the displacement field,

  • \(f\) is the density of body forces,

  • \(g\) is the prescribed surface traction,

  • \(n\) is the outward unit normal vector.

Constitutive Law#

Assuming small deformations and isotropic linear elasticity, the stress tensor follows Hooke’s law

\[\sigma(u) = 2\mu e(u) + \lambda \operatorname{tr}(e(u))I,\]

where

\[e(u) = \frac{1}{2} \left( \nabla u + \nabla u^T \right)\]

is the infinitesimal strain tensor.

The Lamé coefficients are expressed in terms of the Young’s modulus \(E\) and Poisson’s ratio \(\nu\) as

\[\lambda = \frac{\nu E} {(1+\nu)(1-2\nu)},\]
\[\mu = \frac{E} {2(1+\nu)}.\]

Static Buckling#

The static buckling analysis evaluates the stability of a structure subjected to compressive loading and determines the critical load factor associated with the onset of instability.

Governing Equations#

The analysis is performed in two stages.

First, the static equilibrium problem is solved to obtain the prestressed configuration.

Then, the critical load factors are obtained by solving the generalized eigenvalue problem

\[\left( K + \lambda K_G \right)u = 0,\]

where

  • \(K\) is the elastic stiffness matrix,

  • \(K_G\) is the geometric stiffness matrix,

  • \(\lambda\) is the critical load factor,

  • \(u\) is the buckling mode.

OpenPisco Implementations#

For a detailed description of the formulation and numerical treatment, refer to the dedicated theoretical reference 1.

Harmonic Analysis#

Harmonic analysis predicts the steady-state response of a structure subjected to periodic loading.

Governing Equations#

We consider the harmonic response problem of a visco-elastic body depicted by the following problem

\[\begin{split}\begin{equation} \label{u} \left\{ \begin{array}{rlll} \rho \ddot{u} + c\dot{u} - \text{div}(\sigma (u)) &= & 0 & \text{in }\Omega \times \mathbb{R}^{+}, \\ u &= & 0 & \text{on }\Gamma_D\times \mathbb{R}^{+},\\ \sigma(u) \cdot n &= &g(t) & \text{on }\Gamma_N\times \mathbb{R}^{+},\\ \sigma(u) \cdot n &= &0 & \text{on }\Gamma_R \cup \Gamma_M \times\mathbb{R}^{+}. \end{array} \right. \end{equation}\end{split}\]

where:

  • \(\sigma\) is the stress tensor satisfying the Hooke’s law

  • \(\rho\) is the material density

  • \(c\) is the damping operator

  • \(g\) is the density of surfaces forces acting on \(\Gamma_N\)

  • the body is clamped on \(\Gamma_D\)