Boroomand et al 2004a - Revision history

Cinmemj: Cinmemj moved page Draft Samper 896389445 to Boroomand et al 2004a

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Cinmemj moved page Draft Samper 896389445 to Boroomand et al 2004a

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Cinmemj at 11:46, 18 June 2018

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Cinmemj at 10:57, 18 June 2018

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@@ Line 368: / Line 368: @@
 Having found the base functions,''' '''the derivatives with respect to the global coordinates may be computed by the chain rule
- [[Image:draft_Samper_896389445-image68.png|72px]] ,             [[Image:draft_Samper_896389445-image69.png|72px]] (17a)
+{| class="formulaSCP" style="width: 100%; text-align: center;"
 or
- [[Image:draft_Samper_896389445-image70.png|84px]] ,         [[Image:draft_Samper_896389445-image71.png|114px]] ,              [[Image:draft_Samper_896389445-image72.png|84px]] (17b)
+{| class="formulaSCP" style="width: 100%; text-align: center;"
 and so forth.  The effect of such a mapping is illustrated in the following sample problem.

@@ Line 617: / Line 617: @@
 One can use the following equation to replace  [[Image:draft_Samper_896389445-image146.png|12px]] by  [[Image:draft_Samper_896389445-image147.png|12px]]
 {|
 |-
 | [[Image:draft_Samper_896389445-image148.png|center|54px]]
 |}
-(29)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (29)
 with  [[Image:draft_Samper_896389445-image149.png|12px]] being the following transformation matrix

@@ Line 106: / Line 106: @@
 where  [[Image:draft_Samper_896389445-image12.png|12px]] is the number of nodes selected in the vicinity of the  [[Image:draft_Samper_896389445-image13.png|6px]] -th master node.  Obviously, to obtain a square system of equations,  [[Image:draft_Samper_896389445-image14.png|12px]] must, at least, be equal to the number of monomials in (3). Unfortunately, the application of the method on some grids may lead to a singular or ill-conditioned system of equations. Some of such arrangements are shown in Figure 1.  To tackle the problem, a least square procedure with the number of nodal unknowns greater than the number of monomials is usually employed. However even in such least square fitting, there is not a guarantee to avoid an ill-conditioned matrix. A detailed description for this class of approximation methods is given in the next section.
 ==2.2 Weighted Least Square Method==
@@ Line 126: / Line 136: @@
 where  [[Image:draft_Samper_896389445-image22.png|24px]] is a vector of base monomials and  [[Image:draft_Samper_896389445-image23.png|12px]] is a vector of coefficients.
 As mentioned in the previous section, the values of  [[Image:draft_Samper_896389445-image24.png|12px]] for the  [[Image:draft_Samper_896389445-image25.png|6px]] -th cloud are determined from the satisfaction of (4). Assuming that generally  [[Image:draft_Samper_896389445-image26.png|12px]] is greater than  [[Image:draft_Samper_896389445-image27.png|12px]] , then satisfaction of (4) requires a least square procedure, which leads to minimization of the following discrete norm

← Older revision		Revision as of 11:07, 18 June 2018
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	[33] Timoshenko S. P. and Goodier J.N., ''“Theory of Elasticity”'' (3<sup>rd</sup> edn.) McGraw-Hill, New York, 1970.		[33] Timoshenko S. P. and Goodier J.N., ''“Theory of Elasticity”'' (3<sup>rd</sup> edn.) McGraw-Hill, New York, 1970.
−
−	~~==List of tables==~~
−
−	~~1 Values of shape function at a master node with non-mapped and mapped clouds.~~
−
−	~~----~~
−
−	~~<span id="fn-1"></span>([[#fnc-1\|<sup>1</sup>]]) Associate Professor in Civil Engineering (IUT), Email: [mailto:boromand@cc.iut.ac.ir boromand@cc.iut.ac.ir]~~
−
−	~~<span id="fn-2"></span>([[#fnc-2\|<sup>2</sup>]]) Graduate student~~
−
−	~~<span id="fn-3"></span>([[#fnc-3\|<sup>3</sup>]]) Professor in Civil Engineering (CIMNE)~~

← Older revision		Revision as of 11:06, 18 June 2018
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	[33] Timoshenko S. P. and Goodier J.N., ''“Theory of Elasticity”'' (3<sup>rd</sup> edn.) McGraw-Hill, New York, 1970.		[33] Timoshenko S. P. and Goodier J.N., ''“Theory of Elasticity”'' (3<sup>rd</sup> edn.) McGraw-Hill, New York, 1970.
−
−	~~==List of figures==~~
−
−	~~''' '''1 (a) Singular arrangement of nodes for fitting a quadratic polynomial; (b) Ill-conditioned~~
−
−	~~arrangement of nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30~~
−
−	2 (a) Domain [[Image:draft_Samper_896389445-image16.png\|12px]] with [[Image:draft_Samper_896389445-image282.png\|6px]] nodes; (b) Sub-domain [[Image:draft_Samper_896389445-image283.png\|18px]] with [[Image:draft_Samper_896389445-image284.png\|12px]] nodes. . . . . . . . . . . . . . . . 30
−
−	~~3 Normalized Gaussian weight function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31~~
−
−	~~4 Cloud selection for a typical master node: (a) Isotropic distribution of nodes;~~
−
−	~~(b) Non-isotropic distribution of nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31~~
−
−	~~5 (a) Non-isotropic grid; (b) Cloud of a typical node; (c) Normalized cloud of the typical node. . 32~~
−
−	~~6 (a) A primary circular cloud with [[Image:draft_Samper_896389445-image285.png\|12px]] points, (b) Main cloud is built through finding the~~
−
−	~~principal axes and proper aspect ratio, (c) normalized cloud with circular weight support. . . . 32~~
−
−	~~7 Evaluation of [[Image:draft_Samper_896389445-image286.png\|12px]] and [[Image:draft_Samper_896389445-image287.png\|12px]] values for Neumann boundary nodes: point 1 is the closest~~
−
−	inside point to master node [[Image:draft_Samper_896389445-image288.png\|6px]] , [[Image:draft_Samper_896389445-image289.png\|6px]] and [[Image:draft_Samper_896389445-image290.png\|12px]] are the projections of constructed sub-domains [[Image:draft_Samper_896389445-image291.png\|12px]]
−
−	and [[Image:draft_Samper_896389445-image292.png\|12px]] to the boundary of the domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
−
−	~~8 (a) Simply supported beam under uniform load; (b) skew grid; (c) Cartesian grid. . . . . . . . 34~~
−
−	~~9 Vertical displacement at the top fiber of simply supported beam. . . . . . . . . . . . . . . . 34~~
−
−	~~10 Solution of simply supported beam under uniform load using conventional and stabilized~~
−
−	~~versions of FPM; (a) Error of maximum vertical displacement; (b) Error of maximum~~
−
−	~~tensile stress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35~~
−
−	~~11 Simply supported beam under uniform load: (a) Distribution of vertical displacement~~
−
−	~~along central fiber of the beam (y=0.5); (b) Distribution of normal stress along the central~~
−
−	~~cross section of the beam (x=4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36~~
−
−	~~12 Laplace equation in a square domain: (a) The domain; (b) Solutions by conventional and~~
−
−	~~stabilized versions of FPM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37~~
−
−	~~13 Cantilever beam under unit end load. Solution with conventional FPM; (a) geometry;~~
−
−	~~(b) Error norms for each component of displacement field; (c) Error norms for each~~
−
−	~~component of stress field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38~~
−
−	~~14 Cantilever beam under unit end load. Solution with stabilized version of FPM;~~
−
−	~~(a) Error norms for components of displacement field; (b) Error norms for components~~
−
−	~~of stress field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39~~
−
−	~~15 Cantilever beam under unit and load; (a) Distribution of vertical displacement along~~
−
−	~~central fiber of the beam (y=0.5); (b) Distribution of shear stress along the central cross~~
−
−	~~section of the beam (x=2.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40~~
−
−	~~16 Square plate under tension: (a) Infinite domain under tension; (b) A quarter of the finite~~
−
−	~~domain under tractions; (c) A grid of nodes for solution of the problem. . . . . . . . . . . . 41~~
−
−	~~17 Stabilized FPM solution of square plate with hole: (a) Error norms for components of~~
−
−	~~displacement field; (b) Error norms for components of stress field; (c) Distribution of~~
−
−	~~vertical displacement along line x=0; (d) Distribution of horizontal stress along line x=0. . . . 42~~

	==List of tables==		==List of tables==

← Older revision	Revision as of 12:40, 18 June 2018
(No difference)

@@ Line 1,090: / Line 1,090: @@
 | [[Image:draft_Samper_896389445-image273.png|300px]]
 |-
-| [[Image:draft_Samper_896389445-image274.png|center|294px]]
+| [[Image:draft_Samper_896389445-image274.png|294px]]
 |-
 | [[Image:draft_Samper_896389445-image275.png|288px]]
 |-
-| [[Image:draft_Samper_896389445-image276.png|center|396px]]
+| [[Image:draft_Samper_896389445-image276.png|396px]]
 |-
 |[[Image:draft_Samper_896389445-image277.png|216px]]

@@ Line 704: / Line 704: @@
-where  <math>\boldsymbol{\cal A}</math> and  <math>\bf{\cal B}</math> are differential operators for the governing equation in the domain and the Neuman boundary conditions, respectively. The domain  [[Image:draft_Samper_896389445-image16.png|12px]] is a subset of  [[Image:draft_Samper_896389445-image169.png|18px]] in which equation (37) is to be satisfied and  [[Image:draft_Samper_896389445-image170.png|66px]] is the boundary of the domain on which equations (38a) and (38b) are to be satisfied.
+where  <math>\boldsymbol{\cal A}</math> and  <math>\boldsymbol{\cal B}</math> are differential operators for the governing equation in the domain and the Neuman boundary conditions, respectively. The domain  [[Image:draft_Samper_896389445-image16.png|12px]] is a subset of  [[Image:draft_Samper_896389445-image169.png|18px]] in which equation (37) is to be satisfied and  [[Image:draft_Samper_896389445-image170.png|66px]] is the boundary of the domain on which equations (38a) and (38b) are to be satisfied.
 To solve the problem numerically, we start from the general weighted residual expression.  Thus
@@ Line 749: / Line 749: @@
 ''Weak formulation''
-The weak formulation is obtained when an integral by parts is performed of the integral containing the operator  <math>\cal A</math> giving
+The weak formulation is obtained when an integral by parts is performed of the integral containing the operator  <math>\boldsymbol{\cal A}</math> giving
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 762: / Line 762: @@
-Where  <math>\cal A_1</math>,  <math>\cal A_2</math> and  <math>\cal C</math> are suitable operators extracted from the integration by part and consistent with operator  <math>\cal A</math>, for instance  <math>\cal A\equiv \cal A_1 \cal A_2</math>.  In boundary value problems, usually, operators  <math>\cal B</math> and  <math>\cal C</math> contain similar terms so that they can be eliminated when the weighting functions are selected by choosing
+Where  <math>\boldsymbol{\cal A}_1</math>,  <math>\boldsymbol{\cal A}_2</math> and  <math>\boldsymbol{\cal C}</math> are suitable operators extracted from the integration by part and consistent with operator  <math>\boldsymbol{\cal A}</math>, for instance  <math>\boldsymbol{\cal A}\equiv \boldsymbol{\cal A}_1 \boldsymbol{\cal A}_2</math>.  In boundary value problems, usually, operators  <math>\boldsymbol{\cal B}</math> and  <math>\boldsymbol{\cal C}</math> contain similar terms so that they can be eliminated when the weighting functions are selected by choosing
 {| class="formulaSCP" style="width: 100%; text-align: center;"