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Register a new userA General-Purpose, Inelastic, Rotation-Free Kirchhoff-Love Shell Formulation for Peridynamics
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Added by
Masoud Behzadinasab
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We present a comprehensive rotation-free Kirchhoff-Love (KL) shell formulation for peridynamics (PD) that is capable of modeling large elasto-plastic deformations and fracture in thin-walled structures. To remove the need for a predefined global parametric domain, Principal Component Analysis is employed in a meshfree setting to develop a local parameterization of the shell midsurface. The KL shell kinematics is utilized to develop a correspondence-based PD formulation. A bond-stabilization technique is employed to naturally achieve stability of the discrete solution. Only the mid-surface velocity degrees of freedom are used in the governing thin-shell equations. 3D rate-form material models are employed to enable simulating a wide range of material behavior. A bond-associative damage correspondence modeling approach is adopted to use classical failure criteria at the bond level, which readily enables the simulation of brittle and ductile fracture. NAT{Discretizing the model with asymptotically compatible meshfree approximation provides a scheme which converges to the classical KL shell model while providing an accurate and flexible framework for treating fracture.} A wide range of numerical examples, ranging from elastostatics to problems involving plasticity, fracture, and fragmentation, are conducted to validate the accuracy, convergence, and robustness of the developed PD thin-shell formulation. It is also worth noting that the present method naturally enables the discretization of a shell theory requiring higher-order smoothness on a completely unstructured surface mesh.
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Stable and accurate modeling of thin shells requires proper enforcement of all types of boundary conditions. Unfortunately, for Kirchhoff-Love shells, strong enforcement of Dirichlet boundary conditions is difficult because both functional and derivative boundary conditions must be applied. A popular alternative is to employ Nitsches method to weakly enforce all boundary conditions. However, while many Nitsche-based formulations have been proposed in the literature, they lack comprehensive error analyses and verifications. In fact, existing formulations are variationally inconsistent and yield sub-optimal convergence rates when used with common boundary condition specifications. In this paper, we present a novel Nitsche-based formulation for the linear Kirchhoff-Love shell that is provably stable and optimally convergent for general sets of admissible boundary conditions. To arrive at our formulation, we first present a framework for constructing Nitsches method for any abstract variational constrained minimization problem. We then apply this framework to the linear Kirchhoff-Love shell and, for the particular case of NURBS-based isogeometric analysis, we prove that the resulting formulation yields optimal convergence rates in both the shell energy norm and the standard $L^2$-norm. In the process, we derive the Euler-Lagrange equations for general sets of admissible boundary conditions and show that the Euler-Lagrange boundary conditions typically presented in the literature is incorrect. We verify our formulation by manufacturing solutions for a new shell obstacle course that encompasses flat, parabolic, hyperbolic, and elliptic geometric configurations. These manufactured solutions allow us to robustly measure the error across the entire shell in contrast with current best practices where displacement and stress errors are only measured at specific locations.
A method to simulate orthotropic behaviour in thin shell finite elements is proposed. The approach is based on the transformation of shape function derivatives, resulting in a new orthogonal basis aligned to a specified preferred direction for all elements. This transformation is carried out solely in the undeformed state leaving minimal additional impact on the computational effort expended to simulate orthotropic materials compared to isotropic, resulting in a straightforward and highly efficient implementation. This method is implemented for rotation-free triangular shells using the finite element framework built on the Kirchhoff--Love theory employing subdivision surfaces. The accuracy of this approach is demonstrated using the deformation of a pinched hemispherical shell (with a 18{deg} hole) standard benchmark. To showcase the efficiency of this implementation, the wrinkling of orthotropic sheets under shear displacement is analyzed. It is found that orthotropic subdivision shells are able to capture the wrinkling behavior of sheets accurately for coarse meshes without the use of an additional wrinkling model.
We develop and analyze an ultraweak variational formulation for a variant of the Kirchhoff-Love plate bending model. Based on this formulation, we introduce a discretization of the discontinuous Petrov-Galerkin type with optimal test functions (DPG). We prove well-posedness of the ultraweak formulation and quasi-optimal convergence of the DPG scheme. The variational formulation and its analysis require tools that control traces and jumps in $H^2$ (standard Sobolev space of scalar functions) and $H(mathrm{div,Div})$ (symmetric tensor functions with $L_2$-components whose twice iterated divergence is in $L_2$), and their dualities. These tools are developed in two and three spatial dimensions. One specific result concerns localized traces in a dense subspace of $H(mathrm{div,Div})$. They are essential to construct basis functions for an approximation of $H(mathrm{div,Div})$. To illustrate the theory we construct basis functions of the lowest order and perform numerical experiments for a smooth and a singular model solution. They confirm the expected convergence behavior of the DPG method both for uniform and adaptively refined meshes.
We extend the analysis and discretization of the Kirchhoff-Love plate bending problem from [T. Fuhrer, N. Heuer, A.H. Niemi, An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation, arXiv:1805.07835, 2018] in two aspects. First, we present a well-posed formulation and quasi-optimal DPG discretization that includes the gradient of the deflection. Second, we construct Fortin operators that prove the well-posedness and quasi-optimal convergence of lowest-order discrete schemes with approximated test functions for both formulations. Our results apply to the case of non-convex polygonal plates where shear forces can be less than $L_2$-regular. Numerical results illustrate expected convergence orders.
In this work, we propose and develop efficient and accurate numerical methods for solving the Kirchhoff-Love plate model in domains with complex geometries. The algorithms proposed here employ curvilinear finite-difference methods for spatial discretization of the governing PDEs on general composite overlapping grids. The coupling of different components of the composite overlapping grid is through numerical interpolations. However, interpolations introduce perturbation to the finite-difference discretization, which causes numerical instability for time-stepping schemes used to advance the resulted semi-discrete system. To address the instability, we propose to add a fourth-order hyper-dissipation to the spatially discretized system to stabilize its time integration; this additional dissipation term captures the essential upwinding effect of the original upwind scheme. The investigation of strategies for incorporating the upwind dissipation term into several time-stepping schemes (both explicit and implicit) leads to the development of four novel algorithms. For each algorithm, formulas for determining a stable time step and a sufficient dissipation coefficient on curvilinear grids are derived by performing a local Fourier analysis. Quadratic eigenvalue problems for a simplified model plate in 1D domain are considered to reveal the weak instability due to the presence of interpolating equations in the spatial discretization. This model problem is further investigated for the stabilization effects of the proposed algorithms. Carefully designed numerical experiments are carried out to validate the accuracy and stability of the proposed algorithms, followed by two benchmark problems to demonstrate the capability and efficiency of our approach for solving realistic applications. Results that concern the performance of the proposed algorithms are also presented.
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