A Study on the LATIN-PGD Method: Analysis of Some Variants in the Light of the Latest Developments

Abstract

The LATIN-PGD method is a powerful alternative to the Newton–Raphson scheme for solving non-linear time-dependent problems in combination with reduced-order modeling methods. Many developments have been carried out over the last few decades and have led to some variants of the LATIN-PGD method. However, only few comparisons have been made between these variants and none using the digital resources now available. In this article, a comparison between the two major variants of the LATIN-PGD method, as well as with the Newton–Raphson one, is performed using a unified software which highlights the assets of the LATIN-PGD. Various test cases dealing with elasto-visco-plastic problems are undertaken, including comparison with commercial solvers, which reveals the interesting time saving in favor of the LATIN-PGD method.

Appendix: Major Contributors to the Developments of the LATIN Method

Scope	LATIN version	Search directions		References	Features	MS	MP	PGD
		${\varvec{\Theta }}^{+}$	${\varvec{\Theta }}^{-}$
Large deformation	${\mathcal {F}}$	$\infty$	$\mathrm {T}_{\widehat{\varvec{s}}}\,( {\Gamma })$	[27]	Plasticity			$\checkmark$
		$\alpha \varvec{{\mathcal {K}}}$	$\alpha \varvec{{\mathcal {K}}}$	[68]	Forming process			$\checkmark$
		$\infty$	$\alpha \varvec{{\mathcal {K}}}$	[1]	Contact friction, sheet cutting			$\checkmark$
				[9]	Visco-plasticity, damage			$\checkmark$
				[10]	Creep damage			$\checkmark$
	$\mathcal {IV}$	$\infty$	$\alpha \varvec{{\mathcal {K}}}$	[24]	Beam buckling			$\checkmark$
				[8]	Elastomers damage
Elasto-visco-plasticity, large number of cycles, fatigue, damage, quasi-brittle materials	${\mathcal {F}}$	$\mathrm {T}_{\widehat{\varvec{s}}}\,( {\Gamma })$	$\mathrm {T}_{\widehat{\varvec{s}}}\,( {\Gamma })$	[62]	Reference work			$\checkmark$
				[17, 18]				$\checkmark$
				[39]	$\mathcal {VI}$ premises			$\checkmark$
				[51]				$\checkmark$
		$\infty$	$\alpha \varvec{{\mathcal {K}}}$	[90]	Snap-backs
	$\mathcal {IV}$	$\mathrm {T}_{\widehat{\varvec{s}}}\,( {\Gamma })$	$\mathrm {T}_{\widehat{\varvec{s}}}\,( {\Gamma })$	[7, 37]				$\checkmark$
				[56]	Reference work			$\checkmark$
				[61]	Generalized variables
		$\infty$	$\mathrm {T}_{\widehat{\varvec{s}}}\,( {\Gamma })$	[23]	Buckling		$\checkmark$	$\checkmark$
				[38]	Thermo-mechanical			$\checkmark$
				[80]	Verification			$\checkmark$
				[82]				$\checkmark$
				[75]			$\checkmark$	$\checkmark$
				[11]	Cyclic damage			$\checkmark$
		$\varvec{{\widehat{{\varvec{\varepsilon }}}}} = \varvec{{\varvec{\varepsilon }}}_{n_{\ell }}$	$\mathrm {T}_{\widehat{\varvec{s}}}\,( {\Gamma })$	[71]	Multi-fidelity kriging		$\checkmark$	$\checkmark$
		$\alpha \varvec{{\mathcal {K}}}$	$\alpha \varvec{{\mathcal {K}}}$	[92]	Concrete damage		$\checkmark$	$\checkmark$
	$\mathcal {DDM}$			[56]
				[79]	MS in space & time	$\checkmark$		$\checkmark$
Thermal	${\mathcal {F}}$	$\mathrm {T}_{\widehat{\varvec{s}}}\,( {\Gamma })$	$\mathrm {T}_{\widehat{\varvec{s}}}\,( {\Gamma })$	[50]	Optimal MP		$\checkmark$	$\checkmark$
Non-linear dynamics, fast dynamics, shocks, wave propagation	${\mathcal {F}}$	$\varvec{{\mathcal {K}}}$	$\varvec{{\mathcal {K}}}$	[85]				$\checkmark$
		$\infty$	$\varvec{{\mathcal {K}}}$	[52]	Sheet cutting			$\checkmark$
		$\infty$	$\mathrm {T}_{\widehat{\varvec{s}}}\,( {\Gamma })$	[47]	Causality principle
	$\mathcal {IV}$	$\infty$	$\mathrm {T}_{\widehat{\varvec{s}}}\,( {\Gamma })$	[43]	Visco-plasticity			$\checkmark$
	$\mathcal {DDM}$			[67]	Assembly
				[22]
				[87]	Composite damage
				[77]		$\checkmark$	$\checkmark$
				[25]	Frictional contact	$\checkmark$	$\checkmark$

Composite damage	${\mathcal {F}}$	$\infty$	$\varvec{{\mathcal {K}}}$	[5]				$\checkmark$
				[4]				$\checkmark$
				[49]
	$\mathcal {IV}$	$\infty$	$\mathrm {T}_{\widehat{\varvec{s}}}\,( {\Gamma })$	[41]
	$\mathcal {DDM}$			[53, 89, 91]		$\checkmark$
				[84]	Assembly	$\checkmark$	$\checkmark$
				[86]	Buckling	$\checkmark$
Inverse problems, identification	$\mathcal {IV}$	$\infty$	$\mathrm {T}_{\widehat{\varvec{s}}}\,( {\Gamma })$	[6]				$\checkmark$
				[73]	Visco-plasticity, damage
Assembly, frictional contact, parallelism, cracking	$\mathcal {DDM}$			[63]
				[16]	2D-axisymmetrical
				[36]
				[33]	3D
				[57]		$\checkmark$
				[15]
				[58]
				[69]	Composite	$\checkmark$
				[21]			$\checkmark$
				[74]	Visco-plasticity, damage	$\checkmark$
				[20]	Optimization
				[48]	XFem	$\checkmark$
				[3]	Discrete systems	$\checkmark$
				[28]	Damping	$\checkmark$
				[45]	POD & MG-FAS		$\checkmark$
				[29]	Hyper-reduction RPM	$\checkmark$		$\checkmark$
				[78]	Non-intrusive	$\checkmark$
Multi-physics	$\mathcal {DDM}$			[42, 76]	Poro-elasticity	$\checkmark$		$\checkmark$

1. MS multi-scale, MP multi-parametric, $\mathrm {T}_{\widehat{\varvec{s}}}\,({\Gamma })$ search direction related to tangential operator