In this contribution we address the issue of efficient finite element treatment for phase-field modeling of brittle fracture. We start by providing an overview of the existing quasi-static and dynamic phase-field fracture formulations from the physics and the mechanics communities. Within the formulations stemming from Griffith's theory, we focus on quasi-static models featuring a tension-compression split, which prevent cracking in compression and interpenetration of the crack faces upon closure, and on the staggered algorithmic implementation due to its proved robustness. In this paper, we establish an appropriate stopping criterion for the staggered scheme. Moreover, we propose and test the so-called hybrid formulation, which leads within a staggered implementation to an incrementally linear problem. This enables a significant reduction of computational cost--about one order of magnitude--with respect to the available (non-linear) models. The conceptual and structural similarities of the hybrid formulation to gradient-enhanced continuum damage mechanics are outlined as well. Several benchmark problems are solved, including one with own experimental verification.

A review on phase-field models of brittle fracture and a new fast hybrid formulation

Phase-field modeling of brittle fracture in elastic solids is a well-established framework that overcomes the limitations of the classical Griffith theory in the prediction of crack nucleation and in the identification of complicated crack paths including branching and merging. We propose a novel phase-field model for ductile fracture of elasto-plastic solids in the quasi-static kinematically linear regime. The formulation is shown to capture the entire range of behavior of a ductile material exhibiting $$J_2$$J2-plasticity, encompassing plasticization, crack initiation, propagation and failure. Several examples demonstrate the ability of the model to reproduce some important phenomenological features of ductile fracture as reported in the experimental literature.

http://congress.cimne.com/coupled2015/admin/files/fileabstract/a482.pdf

Phase-field modeling of ductile fracture

A variational free-discontinuity formulation of brittle fracture was given by Francfortand Marigo [1], where the total energy is minimized with respect to the crackgeometry and the displacement field simultaneously. The entire evolution of cracksincluding their initiation and branching is determined by this minimization principlerequiring no further criterion. However, a direct numerical discretization of themodel faces considerable difficulties as the displacement field is discontinuous inthe presence of cracks.

A Continuum Phase Field Model for Fracture

Fracture is one of the most commonly encountered failure modes of engineering materials and structures. Prevention of cracking-induced failure is, therefore, a major concern in structural designs. Computational modeling of fracture constitutes an indispensable tool not only to predict the failure of cracking structures but also to shed insights into understanding the fracture processes of many materials such as concrete, rock, ceramic, metals, and biological soft tissues. This chapter provides an extensive overview of the literature on the so-called phase-field fracture/damage models (PFMs), particularly, for quasi-static and dynamic fracture of brittle and quasi-brittle materials, from the points of view of a computational mechanician. PFMs are the regularized versions of the variational approach to fracture which generalizes Griffith's theory for brittle fracture. They can handle topologically complex fractures such as initiation, intersecting, and branching cracks in both two and three dimensions with a quite straightforward implementation. One of our aims is to justify the gaining popularity of PFMs. To this end, both theoretical and computational aspects are discussed and extensive benchmark problems (for quasi-static and dynamic brittle/cohesive fracture) that are successfully and unsuccessfully solved with PFMs are presented. Unresolved issues for further investigations are also documented.

Phase-field modeling of fracture

Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model

Numerical methods that are able to predict the failure of technical structures due to fracture are important in many engineering applications. One of these approaches, the so-called phase field method, represents cracks by means of an additional continuous field variable. This strategy avoids some of the main drawbacks of a sharp interface description of cracks. For example, it is not necessary to track or model crack faces explicitly, which allows a simple algorithmic treatment. The phase field model for brittle fracture presented in Kuhn and Muller (Eng Fract Mech 77(18):3625---3634, 2010) assumes quasi-static loading conditions. However dynamic effects have a great impact on the crack growth in many practical applications. Therefore this investigation presents an extension of the quasi-static phase field model for fracture from Kuhn and Muller (Eng Fract Mech 77(18):3625---3634, 2010) to the dynamic case. First of all Hamilton's principle is applied to derive a coupled set of Euler-Lagrange equations that govern the mechanical behaviour of the body as well as the crack growth. Subsequently the model is implemented in a finite element scheme which allows to solve several test problems numerically. The numerical examples illustrate the capabilities of the developed approach to dynamic fracture in brittle materials.

Phase field approximation of dynamic brittle fracture

Signed-digit (SD) numbers generalize traditional radix numbers by allowing negative digits within a certain range. Typically, this leads to redundant number representations that can be used to avoid the carry propagation problem of addition of radix numbers. Unfortunately, as proved by Avizienis, the standard algorithm for carry-free addition of SD numbers does not work for the binary case. In this paper, we therefore construct a special algorithm for the carry-free addition and subtraction of binary SD numbers, i.e., addition and subtraction of n-digit numbers are performed with circuits of depth O(1) and size O(n). This is possible by computing in addition to the transfer digits used by the standard algorithm one additional bit that allows us to distinguish relevant cases to avoid propagation of dependencies. The additional bit and the transfer digit used to compute the sum digit at position i depend only on the summands' digits at positions i and i -- 1 so that all sum digits can be computed with a hardware circuit of a depth that is independent of the number of digits. We first explain the basics of the standard addition algorithm to derive the additional information needed to fix the algorithm for the binary case. After proving the correctness of our algorithm, we present experimental results that show that our implementation clearly outperforms two's complement addition even for small numbers, and saves 50% of the required chip area compared to other carry-free implementations.

https://es.cs.uni-kl.de/publications/datarsg/ScWi14.pdf

A New Algorithm for Carry-Free Addition of Binary Signed-Digit Numbers

Synchronous programs offer comfortable statements for preemption, which allow statements to abort or suspend other statements. However, while these preemption statements can be used to conveniently describe complex control behaviors in a concise, but yet precise way, they impose very difficult problems for the modular synthesis of systems. For this reason, the integration and re-use of already pre-compiled IP cores has so far only been possible in two restricted variants: either by calls to instantaneous procedures of the host language or by calls to asynchronous tasks of the host language. The first variant is restricted to combinational circuits, and the second variant breaks the synchronous design

Using IP Cores in Synchronous Languages.

In this paper, we present a coprocessor to accelerate explicit state based model checking by computing the set of reachable states with the help of massively parallel hardware. The algorithm is based on known implementations of algorithms for the solution of the algebraic path problem on systolic arrays. We describe the mapping to a field programmable gate array, including the implementation of input and output operations. Our preliminary experimental results show a significant speedup and the applicability of our approach to real-world problems.

https://es.cs.uni-kl.de/publications/datarsg/BrSW08.pdf

Adrian Willenbucher

Papers

Phase field approximation of dynamic brittle fracture

A New Algorithm for Carry-Free Addition of Binary Signed-Digit Numbers

Using IP Cores in Synchronous Languages.

Hardware Acceleration for Model Checking.

GPU-Accelerated Crack Path Computation Based on a Phase Field Approach for Brittle Fracture