Hernandez et al 2014b - Revision history

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← Older revision		Revision as of 15:23, 25 November 2019
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	<span id="fn-11"></span>		<span id="fn-11"></span>
−	<span style="text-align: center; font-size: 75%;">([[#fnc-11\|<sup>1</sup>]]) Incidentally, this way of proceeding has the flavor of the nonlinear model reduction strategy advocated by Carlberg and co-workers <span id='citeF-78'></span><span id='citeF-80'></span>[[#cite-78\|[78,80]], in which the reduction is carried over the linearized form of the pertinent governing equation</span>	+	<span style="text-align: center; font-size: 75%;">([[#fnc-11\|<sup>1</sup>]]) Incidentally, this way of proceeding has the flavor of the nonlinear model reduction strategy advocated by Carlberg and co-workers <span id='citeF-78'></span><span id='citeF-80'></span>[[#cite-78\|[78,80]]], in which the reduction is carried over the linearized form of the pertinent governing equation</span>

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 ===1.1.1 Moore's and Parkinson's laws in computational modeling===
-A basic precept in constructing a successful computational model &#8211;-one that strikes the right balance between  accuracy and  simplicity&#8211;- is, quoting M.Ashby <span id='citeF-1'></span>[[#cite-1|[1]]],  ``to unashamedly distort the ''inessentials'' in order to capture the features that really matter''.  Yet with the general availability of fast computers with large memories, and the  exponential growth   presaged by Moore's law  of such capacities, it is becoming increasingly  difficult to resist  the temptation to   violate this basic precept and,    in the quest for higher accuracy,  indiscriminately include features that do not contribute significantly to the overall response of the modeled system.   As   pointed out by  Venkataraman et al.<span id='citeF-2'></span>[[#cite-2|[2]]],   this  had led to the paradox  that certain  engineering problems   that were   computationally tackled for the first time  40 years  ago  ''appear''  to remain    comparatively costly, even though  the   capacity of computers have  increased one trillion-fold during this period. This apparent  paradox is of course not exclusive of computational physical modeling, but rather     just another manifestation of the  general  adage “software applications will invariably grow to fill up increased computer memory, processing capabilities  and storage space”,  known as  the  ''computerized''     ''Parkinson's law'' <span id='citeF-3'></span>[[#cite-3|[3]]].
+A basic precept in constructing a successful computational model &#8211;-one that strikes the right balance between  accuracy and  simplicity&#8211;- is, quoting M.Ashby <span id='citeF-1'></span>[[#cite-1|[1]]],  ``to unashamedly distort the ''inessentials'' in order to capture the features that really matter.  Yet with the general availability of fast computers with large memories, and the  exponential growth   presaged by Moore's law  of such capacities, it is becoming increasingly  difficult to resist  the temptation to   violate this basic precept and,    in the quest for higher accuracy,  indiscriminately include features that do not contribute significantly to the overall response of the modeled system.   As   pointed out by  Venkataraman et al.<span id='citeF-2'></span>[[#cite-2|[2]]],   this  had led to the paradox  that certain  engineering problems   that were   computationally tackled for the first time  40 years  ago  ''appear''  to remain    comparatively costly, even though  the   capacity of computers have  increased one trillion-fold during this period. This apparent  paradox is of course not exclusive of computational physical modeling, but rather     just another manifestation of the  general  adage “software applications will invariably grow to fill up increased computer memory, processing capabilities  and storage space”,  known as  the  ''computerized''     ''Parkinson's law'' <span id='citeF-3'></span>[[#cite-3|[3]]].
 ===1.1.2 The two-scale homogenization problem===

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 One plausible route for determining a low-dimensional approximation space that embraces both statically admissible and statically inadmissible stresses might be to collect, during the  offline finite element calculations, not only converged stresses,    but also   the  unconverged ones &#8211;-i.e., those generated during the corresponding iterative algorithm&#8211;-, and then perform the POD-based dimensionality reduction over the whole ensemble of snapshots<span id="fnc-11"></span>[[#fn-11|<sup>1</sup>]]. In the present work, however, we pursue an approach  that  precludes the necessity of undertaking this  computationally laborious and in some aspects objectionable &#8211;-there is no guarantee that the span of selected, unconverged stress snapshots covers the entire space of statically inadmissible stresses&#8211;-process. The idea behind the  employed approach was   originally conceived, but  not fully developed,    by the authors   in a recent report <span id='citeF-36'></span>[[#cite-36|[36]]]. Here, the theory underlying such an idea is  further elaborated  and cast into the formalisms of functional analysis.
 ====4.3.3.1 Continuum formulation====
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 |}
 <span id="fn-12"></span>
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 <div id="cite-79"></div>
 '''[[#citeF-79|[79]]]''' Hogben, L. (2006) "Handbook of linear algebra". Chapman & Hall/CRC

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 | style="text-align: center;" | <math> = \displaystyle \sum _{k=1}^{n_{snp}} \left\langle {{\Delta \boldsymbol{u}^{*}}^k},{{\Delta \boldsymbol{u}^{*}}^k} \right\rangle_{L_2(\Omega )}      </math>
 |-
-| style="text-align: center;" | <math> =  \displaystyle \sum _{k=1}^{n_{snp}} \left(\sum _{I=1}^{n} \sum _{J=1}^{n} {\Delta \boldsymbol{u}^{*}}_{I}^{k} {\Delta \boldsymbol{u}^{*}}_{J}^{k}        \left\langle {N_{\uppercase{I}}},{ N_{\uppercase{J}}} \right\rangle_{L_2(\Omega )} \right)      </math>
+| style="text-align: center;" | <math> =  \displaystyle \sum _{k=1}^{n_{snp}} \left(\sum _{I=1}^{n} \sum _{J=1}^{n} {\Delta \boldsymbol{u}^{*}}_{I}^{k} {\Delta \boldsymbol{u}^{*}}_{J}^{k}        \left\langle {N_{{I}}},{ N_{{J}}} \right\rangle_{L_2(\Omega )} \right)      </math>
 |-
 | style="text-align: center;" | <math> = \textrm{tr}( (\boldsymbol{X}_{u}- \boldsymbol{X}_{u}^{*})^T \mathbf{M}(\boldsymbol{X}_{u}-\boldsymbol{X}_{u}^{*})) </math>

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 =Appendix A. Connection between POD and SVD=
-We saw in Section [[#3.1.2.1 Proper Orthogonal Decomposition|3.1.2.1]] that the POD problem for the displacement fluctuations boils down to the solution of the following eigenvalue problem:
+We saw in Section [[#3.1.2 Proper Orthogonal Decomposition|3.1.2]] that the POD problem for the displacement fluctuations boils down to the solution of the following eigenvalue problem:
 <span id="eq-A.1"></span>

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 ===4.3.1 The reduced-order  subspace of statically admissible stresses (''V''<sub>&sigma;</sub><sup>*</sup>)===
-Similarly to the problem addressed in Chapter [[#3 Reduced-order model of the RVE|3]] concerning the    reduced basis for the displacement fluctuations,  the problem of  constructing   a <math display="inline">\mathcal{O}(n_u)</math>-dimensional representation of the stress field reduces, in principle, to finding  a set of orthogonal basis functions <math display="inline">\{ \boldsymbol{\mathit{\Psi}}_1(\mathbf{x}), \boldsymbol{\mathit{\Psi}}_2(\mathbf{x}) \ldots \boldsymbol{\mathit{\Psi}}_{n_{\sigma }}(\mathbf{x})\} </math> (<math display="inline">n_{\sigma }= \mathcal{O}(n_u)</math>)  such that its span accurately    approximates     the set of all possible ''stress solutions'' &#8211;-that is, the set of all ''statically admissible stresses''. Accordingly,    the  procedure to  compute the reduced basis for the stress field would be, ''mutatis mutandis'', formally identical to that explained earlier for the displacement fluctuations.  Firstly,  finite element, stress distributions over the cell are computed for representative, input macro-strain histories (the most practical and somehow consistent choice regarding these strain trajectories   is to   use the same as in the   computation of the  displacement fluctuations snapshots).   Then, the elastic/inelastic dimensionality reduction process set forth in Section [[#3.1.2.3 Elastic/Inelastic reduced basis functions|3.1.2.3]] is applied to the resulting ensemble of stress solutions <math display="inline">\{ \boldsymbol{\sigma }^1(\mathbf{x}),\boldsymbol{\sigma }^2(\mathbf{x}) \ldots  \boldsymbol{\sigma }^{n_{snp}}(\mathbf{x}) \} </math>,   in order to identify both the elastic  and the    ''essential'' inelastic  stress modes.  The space spanned  by these  modes   will be denoted hereafter by <math display="inline">\mathcal{V}_{\sigma }^{*}</math> and termed  the ''reduced-order  subspace of statically admissible stresses'':
+Similarly to the problem addressed in Chapter [[#3 Reduced-order model of the RVE|3]] concerning the    reduced basis for the displacement fluctuations,  the problem of  constructing   a <math display="inline">\mathcal{O}(n_u)</math>-dimensional representation of the stress field reduces, in principle, to finding  a set of orthogonal basis functions <math display="inline">\{ \boldsymbol{\mathit{\Psi}}_1(\mathbf{x}), \boldsymbol{\mathit{\Psi}}_2(\mathbf{x}) \ldots \boldsymbol{\mathit{\Psi}}_{n_{\sigma }}(\mathbf{x})\} </math> (<math display="inline">n_{\sigma }= \mathcal{O}(n_u)</math>)  such that its span accurately    approximates     the set of all possible ''stress solutions'' &#8211;-that is, the set of all ''statically admissible stresses''. Accordingly,    the  procedure to  compute the reduced basis for the stress field would be, ''mutatis mutandis'', formally identical to that explained earlier for the displacement fluctuations.  Firstly,  finite element, stress distributions over the cell are computed for representative, input macro-strain histories (the most practical and somehow consistent choice regarding these strain trajectories   is to   use the same as in the   computation of the  displacement fluctuations snapshots).   Then, the elastic/inelastic dimensionality reduction process set forth in Section [[#3.1.2 Elastic/Inelastic reduced basis functions|3.1.2]] is applied to the resulting ensemble of stress solutions <math display="inline">\{ \boldsymbol{\sigma }^1(\mathbf{x}),\boldsymbol{\sigma }^2(\mathbf{x}) \ldots  \boldsymbol{\sigma }^{n_{snp}}(\mathbf{x}) \} </math>,   in order to identify both the elastic  and the    ''essential'' inelastic  stress modes.  The space spanned  by these  modes   will be denoted hereafter by <math display="inline">\mathcal{V}_{\sigma }^{*}</math> and termed  the ''reduced-order  subspace of statically admissible stresses'':
 <span id="eq-4.11"></span>

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	Two crucial aspects of the integration scheme sketched in the foregoing remains to be addressed, namely, the determination of the vector space (hereafter denoted by <math display="inline">\mathcal{V}_{\sigma }^{apr}</math>) in which the low-dimensional approximation of the stress field should lie in order to obtain an accurate and at the same time well-posed HP-ROM; and the calculation of the optimal location of the ''sampling or integration points'' at which the stress tensor is to be evaluated. Attention here and in the next Section is confined to the aspect related to the stress approximation space, while the discussion of the issue related to the selection of sampling points is deferred to Section [[#4.5 Selection of sampling points\|4.5]].		Two crucial aspects of the integration scheme sketched in the foregoing remains to be addressed, namely, the determination of the vector space (hereafter denoted by <math display="inline">\mathcal{V}_{\sigma }^{apr}</math>) in which the low-dimensional approximation of the stress field should lie in order to obtain an accurate and at the same time well-posed HP-ROM; and the calculation of the optimal location of the ''sampling or integration points'' at which the stress tensor is to be evaluated. Attention here and in the next Section is confined to the aspect related to the stress approximation space, while the discussion of the issue related to the selection of sampling points is deferred to Section [[#4.5 Selection of sampling points\|4.5]].
−
−

	<span id='lb-4.3.1'></span>		<span id='lb-4.3.1'></span>

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 The error measures  displayed previously in  Figures [[#img-5|5]] and [[#img-7|7]] only depend on the outcome of the SVD of the  snapshot matrices; they can be   calculated, thus,  ''before'' actually constructing the reduced-order model.  Error analyses based on such  measures serve  the useful purpose  of providing a  first hint of how many  stress and displacement fluctuations modes     are needed to  satisfactorily replicate the full-order, finite element solution, and thereby,  of prospectively evaluating  the  ''viability'' of the reduced basis approach itself.
-However, these ''a priori ''error estimates do not tell the whole story. Expression  ([[#eq-5.4|5.4]]) for the   stress approximation error presumes that the stress solution at the chosen sampling points is the one provided by the finite element model, thus ignoring   the fact that, actually,  in the reduced-order model, and for the general case of nonlinear, dissipative materials &#8211;-for linear elasticity, the approximation is ''exact'', see Remarks [[#theorem-obs:kkl|2]] and [[#theorem-obs:222|D.1.2]]&#8211;- the  stress  information at such     points  at a given time step is already   polluted   by   truncation (in displacement fluctuations and stresses)  and reconstruction (in stresses) errors originated in    previous time steps.   To quantify the extent to which this amalgam of accumulated errors  affects  the predictions furnished by the HP-ROM, it is necessary  to perform a'' consistency analysis''.
+However, these ''a priori ''error estimates do not tell the whole story. Expression  ([[#eq-5.4|5.4]]) for the   stress approximation error presumes that the stress solution at the chosen sampling points is the one provided by the finite element model, thus ignoring   the fact that, actually,  in the reduced-order model, and for the general case of nonlinear, dissipative materials &#8211;-for linear elasticity, the approximation is ''exact'', see Remarks [[#theorem-obs:kkl|1]] and [[#theorem-obs:222|D.1.2]]&#8211;- the  stress  information at such     points  at a given time step is already   polluted   by   truncation (in displacement fluctuations and stresses)  and reconstruction (in stresses) errors originated in    previous time steps.   To quantify the extent to which this amalgam of accumulated errors  affects  the predictions furnished by the HP-ROM, it is necessary  to perform a'' consistency analysis''.
 Generally speaking, a reduced basis approximation is said to be ''consistent'' if, in the limit of no truncation, it introduces  no additional error in the solution of the same problem for which the data used in constructing the basis functions were acquired <span id='citeF-78'></span>[[#cite-78|[78]]]. In the BVP under consideration, thus, consistency implies  that,  when using as input macro-strain paths the same trajectories employed in the “training” process,  results obtained with the HP-ROM should converge, as <math display="inline">n_{\sigma }</math> and <math display="inline">n_u</math> increases, to the solution   furnished by the full-order, finite element model.