@Article{icces.2023.09052,
AUTHOR = {Yihao Chen, Yongxing Shen},
TITLE = {A	\"Parallel	Universe\"	Scheme	for	Crack	Nucleation	in	the	Phase	Field	 Approach	to	Fracture},
JOURNAL = {The International Conference on Computational \& Experimental Engineering and Sciences},
VOLUME = {26},
YEAR = {2023},
NUMBER = {3},
PAGES = {1--1},
URL = {http://www.techscience.com/icces/v26n3/54049},
ISSN = {1933-2815},
ABSTRACT = {Crack	nucleation	is	crucial	in	many	industrial	applications. The	phase	field	method	for	fracture	transforms	
the	crack	nucleation	problem	into	a	minimization	problem	of	the	sum	of	the	elastic	potential	energy	and	the	
crack	surface	energy.	Due	to	the	polyconvexity	of	the	formulation,	starting	from	a	crackless	solid,	a standard	
Newton	iteration	may	lead	 to	a	solution	with	no	crack,	even	 though	a	cracked	solution	has	a	lower	 total	
energy.	As	such,	the	critical	load	for	cracking	is	highly	overestimated.	Here,	we	propose	an	algorithm	termed	
“parallel	universe” algorithm	to capture	the	global	minimum.	This	algorithm	has	two	key	ingredients:	(a)	a	
necessary	 condition	 for	 cracking	 solely	 based	 on	 the	 current	 crackless	 solution,	and	 (b)	 beginning	 from	
when	this	condition	is	met,	Newton	iteration	with	two	initial	guesses,	a	crackless	one	and	a	cracked	one,	will	
both	be	performed	and	the	converged	candidate	solution	with	lower	energy	is	accepted	as	the	solution	at	
that	load	step.	Once	the	cracked	candidate	solution	is	accepted,	the	crackless	one	is	discarded,	i.e.,	only	one	
universe	is	retained.	This	cracked	initial	guess	is	obtained	only	once	for	all	load	steps	by	solving	a	series	of	
similar	minimization	problems	with	a	progressively	reduced	critical	crack	energy	release	rate.	Numerical	
examples	 with	 isotropic	 and	 anisotropic	 critical	 crack	 energy	 release	 rates	 indicate	 that	 the	 proposed	
algorithm	is	more	reliable	 (as	 there	is	no	need	 to	retrace)	and	more	efficient	 than	 the	standard	Newton	
iteration	and	a	well-known	backtracking	algorithm.},
DOI = {10.32604/icces.2023.09052}
}