[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

Modeling of the evolution of distributed damage such as microcracking, void formation, and softening frictional slip necessitates strain-softening constitutive models. The nonlocal continuum concept has emerged as an effective means for regularizing the boundary value problems with strain softening, capturing the size effects and avoiding spurious localization that gives rise to pathological mesh sensitivity in numerical computations. A great variety of nonlocal models have appeared during the last two decades. This paper reviews the progress in the nonlocal models of integral type, and discusses their physical justifications, advantages, and numerical applications.

Nonlocal integral formulations of plasticity and damage: Survey of progress

The aim of the present chapter is to provide an in-depth survey of arbitrary Lagrangian–Eulerian (ALE) methods, including both conceptual aspects of the mixed kinematical description and numerical implementation details. Applications are discussed in fluid dynamics, nonlinear solid mechanics and coupled problems describing fluid–structure interaction. The need for an adequate mesh-update strategy is underlined, and various automatic mesh-displacement prescription algorithms are reviewed. This includes mesh-regularization methods essentially based on geometrical concepts, as well as mesh-adaptation techniques aimed at optimizing the computational mesh according to some error indicator. Emphasis is then placed on particular issues related to the modeling of compressible and incompressible flow and nonlinear solid mechanics problems. This includes the treatment of convective terms in the conservation equations for mass, momentum, and energy, as well as a discussion of stress-update procedures for materials with history-dependent constitutive behavior.


Keywords:

ALE description;
convective transport;
finite elements;
stabilization techniques;
mesh regularization and adaptation;
fluid dynamics;
nonlinear solid mechanics;
stress-update procedures;
fluid–structure interaction

https://onlinelibrary.wiley.com/doi/pdf/10.1002/0470091355.ecm009

Arbitrary Lagrangian–Eulerian Methods

http://users.auth.gr/~akonsta/2011_Gradient%20elasticity%20in%20statics%20and%20dynamics.pdf

Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results

The softening elastoplastic models present an unsuitable behavior after reaching
the yield strength: unbounded strain localization. Because of the material
instability, which is reflected in the loss of ellipticity of the governing partial
differential equations, the solution depends on the discretization. 
The present work proposes to solve this dependency using the meshless Finite
Points Method. This meshfree spatial discretization technique allows enriching
the governing equations using gradient’s plasticity and introducing an internal
length scale parameter at the material model in order to objectify the solution.

/pdf/a-finite-points-method-approach-for-strain-localization-4uh45apar0.pdf

A finite points method approach for strain localization using the gradient plasticity formulation

In this paper, a well-known numerical benchmark test which is usually solved with displacement control for low values of the load eccentricity is examined for a complete range of eccentricities of the ring load. For a certain range of the eccentricity the response shows either snap-through or snap-back, depending on the controlled variable. Thus, in this range of eccentricities, the test can be used to verify implementations of arc-length algorithms, using the displacement controlled solution as a reference. Moreover, results are presented for large eccentricities beyond the applicability of displacement-controlled strategies.

/pdf/a-note-on-a-numerical-benchmark-test-an-axisymmetric-shell-btt178yfoa.pdf

A note on a numerical benchmark test: an axisymmetric shell under ring loads

Sound transmission through partitions can be modeled as an acoustic fluid–elastic structure interaction problem. The block Gauss–Seidel iterative method is used in order to solve the finite element linear system of equations. The blocks are defined, respecting the fluid and structural domains. The convergence criterion is analyzed and interpreted in physical terms by means of simple one-dimensional problems. This analysis highlights the negative influence on the convergence of a strong degree of coupling between the acoustic domains and the structure. A selective coupling strategy has been developed and applied to problems with strong coupling (e.g. double walls).

https://upcommons.upc.edu/bitstream/handle/2099/7820/acta19.pdf?sequence=1

The block gauss–seidel method in sound transmission problems

Numerical modelling of the radiation efficiency of asymmetrical structures

This paper presents an adaptive strategy for phase-field simulations with transition to fracture. The phase-field equations are solved only in small subdomains around crack tips to determine propagation, while an XFEM discretization is used in the rest of the domain to represent sharp cracks, enabling to use a coarser discretization and therefore reducing the computational cost. Crack-tip subdomains move as cracks propagate in a fully automatic process. The same computational mesh is used during all the simulation, with an $h$-refined approximation in the elements in the crack-tip subdomains. Continuity of the displacement between the refined subdomains and the XFEM region is imposed in weak form via Nitsche's method. The robustness of the strategy is shown for some numerical examples in 2D and 3D, including branching and coalescence tests.

Antonio Rodríguez-Ferran

Papers

A finite points method approach for strain localization using the gradient plasticity formulation

A note on a numerical benchmark test: an axisymmetric shell under ring loads

The block gauss–seidel method in sound transmission problems

Numerical modelling of the radiation efficiency of asymmetrical structures

A combined XFEM phase-field computational model for crack growth without remeshing