A Global Parametric Nonlinear Programming Framework for a Geometrically Exact Inelastic Beam Finite Element
Restricted (Penn State Only)
- Author:
- Lyritsakis, Charilaos
- Graduate Program:
- Civil Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- September 09, 2022
- Committee Members:
- Reuben Kraft, Outside Unit Member
Tong Qiu, Outside Field Member
Gordon Warn, Major Field Member
Kostas Papakonstantinou, Chair, Minor Member & Dissertation Advisor
Patrick Fox, Program Head/Chair
Pinlei Chen, Major Field Member - Keywords:
- geometrically-exact beam
nonlinear programming
homotopy continuation
parametric optimization
finite element analysis
numerical continuation
elastoplasticity - Abstract:
- A novel framework for the structural analysis of planar frames is developed, which combines tools from finite element analysis, mathematical programming and homotopy theory. The core building block in this work is a versatile, high-performance, hybrid beam-column element which is formulated based on nonlinear programming principles. The geometrically-exact kinematic assumptions are adopted, thereby allowing for arbitrarily large displacements and rotations, while inelastic behavior is modeled by assuming a discretization of selected cross-sections into layers or fibers, which can incorporate a multiaxial constitutive law but act independently of neighboring layers. The element is capable of capturing both geometric and material nonlinearities with just one element per structural member while maintaining all the numerically attractive properties pertaining to the structure of the resulting global stiffness matrix. The interaction between axial, shear and flexural effects, especially during inelastic deformation, is accounted for by incorporating a multiaxial constitutive law at the level of cross-section fibers. In particular, a fast and robust return-mapping algorithm is developed which is utilizing the zero transverse normal stress assumption in order to arrive at a reduced stress space formulation. This allows for a more efficient stress update both in terms of memory and computation costs. The implementation assumes a J2 von Mises material with combined isotropic and kinematic hardening. Such reformulation of general three-dimensional or axisymmetric stress update procedures is crucial for beam elements that rely on scarce meshes but higher order quadratures in order to achieve accuracy, since elastoplastic analyses by fibre-discretized elements increases the computational cost considerably. The proposed beam element is embedded in a parametric nonlinear programming framework which is developed to facilitate analysis considerations beyond response to mechanical loading. In particular, we take advantage of the optimization reformulation of the underlying variational structure of the mechanics problem and develop a naturally parameterized nonlinear programming framework which can easily handle parametric investigations, such as design optimization or sensitivity analysis, while also providing the theoretical and numerical tools that ensure global convergence characteristics, provided certain regularity conditions hold. A predictor-corrector type numerical continuation algorithm suited for this framework is also developed which can account for any parameterization and derivative discontinuities along the solution path. To improve the performance of the aforementioned numerical continuation algorithm, a reliable and efficient prediction scheme is proposed that is considerably faster than conventional approaches that utilize second derivative information. Instead, a weighted least squares fitting is carried out at the start of an incremental step, provided that a number of previously converged solution points are stored and are available in memory. The curve generated by this fitting process is capable to emulate the local geometry of the actual solution curve much better since an interpolating condition is not enforced. In addition, by using an appropriate weighting function, we can assign increased weights to solutions closer to the current step. This leads to a versatile scheme that can provide a suite of additional options, as far as general predictor-corrector algorithms are concerned and, more importantly, it can be applied for any problem which can be solved by such procedures. It can be particularly attractive for problems described by dense, non-symmetric Jacobians, especially if coupled with iterative correction schemes such as conjugate gradients. The present work is partitioned into four parts: the core hybrid beam element, the multiaxial constitutive model and return mapping algorithm, the naturally parameterized framework for the hybrid element and, finally, the weighted least squares predictor scheme. To validate the efficacy and accuracy of the procedures associated with each individual part developed in this work, several numerical tests are presented in the thesis.