MeshPerturb MATLAB codes for mesh perturbation and automated pre and post processing of post-bifurcation analyses via COMSOL

Recently, there has been an interest in designing systems that employ buckling bifurcation as a desired mechanism to generate motion and/or form. As buckling has traditionally been considered a failure mode to be avoided, computational tools for predicting the state of the system after buckling are not well developed. For example, modules for post-bifurcation studies are not available in most of the commercial finite element software packages. Herein, we provide MATLAB codes that add the ability to perform post-bifurcation studies to the COMSOL package. This is achieved by implementing mesh perturbations as a scheme to introduce geometric imperfections to the system. Additionally, we provide codes to automate pre/post processing the studies. These codes enable performing (i) post-bifurcation analysis, (ii) studies on sensitivity to mesh imperfections, and (iii) unattended parametric studies that require re-meshing the geometry. 1. General purpose: Post-bifurcation analysis In engineering applications, solving a bifurcation problem commonly arises during design and analysis of structures suspect to buckling failure. Solution to a bifurcation problem has two distinct steps: (i) predicting the onset of bifurcation and (ii) predicting the behavior of the system after the onset of bifurcation. These steps require two distinct sets of predictive tools. Herein, we focus on developing the predictive tools for solving the second step, i.e., for post-bifurcation analysis. As buckling has traditionally been considered a failure mode, the first step of predicting the onset of bifurcation has received more attention. Thus, predictive tools for post-bifurcation analysis of structures are not as well developed as those for predicting the onset. This trend is also reflected in the development of commercially available finite element analysis (FEA) software packages. For example, dedicated stability analysis based modules are available for predicting the onset of bifurcation but modules for postbifurcation analysis are absent from most FEA packages. In the past, this lack of predictive tools has not been a major issue due to the limited engineering interest in post-bifurcated systems. However, recently there has been an interest in designing and building systems that employ buckling bifurcation as a desired mechanism to generate motion and/or form. Post-bifurcation analysis is essential for an accurate prediction of form/motion in these systems. Our software codes add the ability to perform postbifurcation studies to the commercially available COMSOL FEA package. This enables one to systematically perform post-bifurcation analyses that were previously not feasible within COMSOL. [email protected], [email protected]. This document is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. http://creativecommons.org/licenses/by-nc-sa/3.0/ Description of MeshPerturb Software 2 2. Specific goal: Automated mesh perturbation Providing an external perturbation to the system is often necessary for observing buckling bifurcation during numerical simulations. The bifurcation point is characterized by intersection of multiple feasible modes/paths that the system may follow after bifurcation. Out of these paths, one or more may be energetically favorable. In the absence of any imperfection, the system is unable to “see” any of these alternate paths. Therefore, the system may remain on an unfavorable path even after the bifurcation point. Figure 1: Imperfection is essential for bifurcation. (a) A “perfect” column does not buckle even at very high loads. (b) Real column buckles at high loads due to presence of imperfections. For example, during buckling of a column there are two feasible modes at the bifurcation point (i) pure compression and (ii) bending. Bending is energetically favorable beyond the bifurcation point. However, a ‘perfect’ column with no imperfections would not buckle even beyond the bifurcation load (Fig. 1). In reality, this ‘non-buckling beyond bifurcation’ phenomenon is never observed due to the presence of imperfections in a real system. However, such non-buckling are routinely observed in an idealized finite element model. Therefore, an imperfection must be applied to the FEA model to ensure that the system follows the favorable post-bifurcation path. The imperfection may be in the form of (i) geometric/mesh imperfection and/or (ii) imperfection in the applied loads and boundary conditions. Herein, we provide this imperfection via perturbation of the mesh. Figure 2: Mesh perturbation as imperfection that enables bifurcation. (a) Perfect mesh that does not bifurcate. (b) Perturbed mesh that bifurcates (perturbation exaggerated for clarity). A perturbed mesh can be obtained from a perfect mesh by ‘moving’ the mesh points around. Our codes enable one to systematically perform this perturbation process. P