Advances in variational and hemivariational inequalities : theory, numerical analysis, and applications

Highlighting recent advances in variational and hemivariational inequalities with an emphasis on theory, numerical analysis and applications, this volume serves as an indispensable resource to graduate students and researchers interested in the latest results from recognized scholars in this relatively young and rapidly-growing field. Particularly, readers will find that the volume’s results and analysis present valuable insights into the fields of pure and applied mathematics, as well as civil, aeronautical, and mechanical engineering. Researchers and students will find new results on well posedness to stationary and evolutionary inequalities and their rigorous proofs. In addition to results on modeling and abstract problems, the book contains new results on the numerical methods for variational and hemivariational inequalities. Finally, the applications presented illustrate the use of these results in the study of miscellaneous mathematical models which describe the contact between deformable bodies and a foundation. This includes the modelling, the variational and the numerical analysis of the corresponding contact processes. Furthermore, it can be used as supplementary reading material for advanced specialized courses in mathematical modeling for students with a strong background knowledge on nonlinear analysis, numerical analysis, partial differential equations, and mechanics of continua

A stochastic continuum damage model for dynamic fracture analysis of quasi-brittle materials using asynchronous Spacetime Discontinuous Galerkin (aSDG) method

The microstructural design has an essential effect on the fracture response of brittle materials. We present a stochastic bulk damage formulation to model dynamic brittle fracture. This model is compared with a similar interfacial model for homogeneous and heterogeneous materials. The damage models are rate-dependent, and the corresponding damage evolution includes delay effects. The evolution equation specifies the rate at which damage tends to its quasi-static limit. The relaxation time of the model introduces an intrinsic length scale for dynamic fracture and addresses the mesh sensitivity problem of earlier damage models with much less computational efforts. The ordinary differential form of the damage equation makes this remedy quite simple and enables capturing the loading rate sensitivity of strain-stress response. A stochastic field is defined for material cohesion and fracture strength to involve microstructure effects in the proposed formulations. The statistical fields are constructed through the Karhunen-Loeve (KL) method.An advanced asynchronous Spacetime Discontinuous Galerkin (aSDG) method is used to discretize the final system of coupled equations. Local and asynchronous solution process, linear complexity of the solution versus the number of elements, local recovery of balance properties, and high spatial and temporal orders of accuracy are some of the main advantages of the aSDG method.Several numerical examples are presented to demonstrate mesh insensitivity of the method and the effect of boundary conditions on dynamic fracture patterns. It is shown that inhomogeneity greatly differentiates fracture patterns from those of a homogeneous rock, including the location of zones with maximum damage. Moreover, as the correlation length of the random field decreases, fracture patterns resemble angled-cracks observed in compressive rock fracture. The final results show that a stochastic bulk damage model produces more realistic results in comparison with a homogenizes model

A Hierarchical Multiscale Method for Nonlocal Fine-scale Models via Merging Weak Galerkin and VMS Frameworks

Many problems in natural sciences and engineering involve phenomena that possess a wide spectrum of material, spatial and temporal scales. Most often these scales are nonlocal and therefore pose great challenge to current computational techniques. Of specific interest to the objectives of this proposal are advection dominated viscous flows leading to anisotropic turbulence where all the scales are nonlocal and they concurrently interact all across. Another class of problems from mathematical physics that we address involves propagating steep fronts wherein interacting discontinuities in the underlying physical fields challenge the stability of the numerical methods. Modeling such problems raises open mathematical questions: How should the information obtained from a model at one level be incorporated into a model at a different level? And secondly, how should these scales be made to communicate seamlessly if both coarse and fine scales are inherently non-local? Focus of the proposed research is a unifying mathematical framework for the development of robust numerical methods in two areas of Computational Fluid Dynamics (CFD): (i) Methods for anisotropic turbulence that are derived consistently from the Navier-Stokes equations via facilitating two-way coupling between global and local scales, and (i) Methods that have sound variational structures for the modeling of problems with steep gradients and propagating discontinuities. Emphasis is placed throughout on variationally consistent interscale coupling with rigorous treatment of the continuity conditions that are critical for the mathematical and algorithmic stability.NSF - DMS-16-20231, 2016-17Ope

Advanced Computational Engineering

The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical developments such as mixed finite element methods (FEMs) and other nonstandard FEMs like least-squares, nonconforming, and discontinuous Galerkin (dG) FEMs. Various aspects on this plus related topics ranging from order-reduction methods to isogeometric analysis has been discussed amongst the pariticpants form mathematics and engineering for a large range of applications

Dynamic earthquake rupture modelled with an unstructured 3-D spectral element method applied to the 2011 M9 Tohoku earthquake

An important goal of computational seismology is to simulate dynamic earthquake rupture and strong ground motion in realistic models that include crustal heterogeneities and complex fault geometries. To accomplish this, we incorporate dynamic rupture modelling capabilities in a spectral element solver on unstructured meshes, the 3-D open source code SPECFEM3D, and employ state-of-the-art software for the generation of unstructured meshes of hexahedral elements. These tools provide high flexibility in representing fault systems with complex geometries, including faults with branches and non-planar faults. The domain size is extended with progressive mesh coarsening to maintain an accurate resolution of the static field. Our implementation of dynamic rupture does not affect the parallel scalability of the code. We verify our implementation by comparing our results to those of two finite element codes on benchmark problems including branched faults. Finally, we present a preliminary dynamic rupture model of the 2011 Mw 9.0 Tohoku earthquake including a non-planar plate interface with heterogeneous frictional properties and initial stresses. Our simulation reproduces qualitatively the depth-dependent frequency content of the source and the large slip close to the trench observed for this earthquak

Dynamic earthquake rupture modelled with an unstructured 3-D spectral element method applied to the 2011 M9 Tohoku earthquake

An important goal of computational seismology is to simulate dynamic earthquake rupture and strong ground motion in realistic models that include crustal heterogeneities and complex fault geometries. To accomplish this, we incorporate dynamic rupture modelling capabilities in a spectral element solver on unstructured meshes, the 3-D open source code SPECFEM3D, and employ state-of-the-art software for the generation of unstructured meshes of hexahedral elements. These tools provide high flexibility in representing fault systems with complex geometries, including faults with branches and non-planar faults. The domain size is extended with progressive mesh coarsening to maintain an accurate resolution of the static field. Our implementation of dynamic rupture does not affect the parallel scalability of the code. We verify our implementation by comparing our results to those of two finite element codes on benchmark problems including branched faults. Finally, we present a preliminary dynamic rupture model of the 2011 M_w 9.0 Tohoku earthquake including a non-planar plate interface with heterogeneous frictional properties and initial stresses. Our simulation reproduces qualitatively the depth-dependent frequency content of the source and the large slip close to the trench observed for this earthquake

Partially Penalized Immersed Finite Element Methods for Elliptic Interface Problems

This article presents new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries. These IFE methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in IFE functions. With the enhanced stability due to the added penalty, not only these IFE methods can be proven to have the optimal convergence rate in the H1-norm provided that the exact solution has sufficient regularity, but also numerical results indicate that their convergence rates in both the H1-norm and the L2-norm do not deteriorate when the mesh becomes finer which is a shortcoming of the classic IFE methods in some situations. Trace inequalities are established for both linear and bilinear IFE functions that are not only critical for the error analysis of these new IFE methods, but also are of a great potential to be useful in error analysis for other IFE methods

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells