Puigferrat et al 2020a - Revision history

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Cinmemj: Cinmemj moved page Draft Samper 437326896 to Puigferrat et al 2020a

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Cinmemj moved page Draft Samper 437326896 to Puigferrat et al 2020a

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-In ''Computer Methods in Applied Mechanics and Engineering'', Article in Press, 2020<br />
+In ''Computer Methods in Applied Mechanics and Engineering'', Vol. 365, pp. 112984, 2020<br />
 DOI: 10.1016/j.cma.2020.112984
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-Published in ''Computer Methods in Applied Mechanics and Engineering'', Article in Press, 2020<br />
+In ''Computer Methods in Applied Mechanics and Engineering'', Article in Press, 2020<br />
 DOI: 10.1016/j.cma.2020.112984
 <!-- metadata commented in wiki content

← Older revision		Revision as of 08:54, 19 March 2020
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		+	Published in ''Computer Methods in Applied Mechanics and Engineering'', Article in Press, 2020<br />
		+	DOI: 10.1016/j.cma.2020.112984
	<!-- metadata commented in wiki content		<!-- metadata commented in wiki content
	==FIC-FEM formulation for the multidimensional transient advection-diffusion-absorption equation==		==FIC-FEM formulation for the multidimensional transient advection-diffusion-absorption equation==

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 The diffusion introduced by matrix <math display="inline">{\boldsymbol D}_s</math> takes care of the instabilities induced by the irregularity of the triangular mesh near boundaries that develop parabolic layes <span id='citeF-62'></span>[[#cite-62|[62]]].
-===<u>Any other element:</u> Dₛ=[0]===
+=====<u>Any other element:</u> Dₛ=[0]=====
 '''''Shock capturing characteristic length vector''''' '''h'''<sub>s<sub>c</sub></sub>

@@ Line 277: / Line 277: @@
 Matrix <math display="inline">{\boldsymbol D}_s</math> in Eq.[[#eq-12.a|12.a]] is defined for different element types as follows.
-===<u>3-noded triangles and 4-noded tetrahedra</u>:===
+=====<u>3-noded triangles and 4-noded tetrahedra</u>:=====
 <span id="eq-13"></span>

← Older revision	Revision as of 13:48, 11 March 2020
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 The FIC equations are the starting point for deriving different stabilized numerical methods. Combining the FIC equations with the Galerkin FEM leads to the so-called FIC-FEM procedure that has been successfully applied to the solution of many problems in convective transport, fluid and solid mechanics such as advection-diffusion <span id='citeF-40'></span><span id='citeF-42'></span><span id='citeF-43'></span><span id='citeF-51'></span>[[#cite-40|[40,42,43,51]]], diffusion-absorption and Helmholtz <span id='citeF-15'></span>[[#cite-15|[15]]], advection-diffusion-absorption <span id='citeF-52'></span><span id='citeF-55'></span>[[#cite-52|[52,55]]], advection-diffusion-reaction <span id='citeF-61'></span>[[#cite-61|[61]]], incompressible fluid flow <span id='citeF-53'></span><span id='citeF-54'></span><span id='citeF-56'></span><span id='citeF-58'></span><span id='citeF-60'></span>[[#cite-53|[53,54,56,58,60]]], fluid-structure-interaction <span id='citeF-45'></span><span id='citeF-49'></span><span id='citeF-57'></span>[[#cite-45|[45,49,57]]], particle-laden flows and standard and incompressible solid mechanics <span id='citeF-47'></span><span id='citeF-50'></span><span id='citeF-59'></span>[[#cite-47|[47,50,59]]]. The FIC approach has also been applied to the solution of a variety of problems in mechanics using the meshless finite point method  <span id='citeF-39'></span><span id='citeF-41'></span><span id='citeF-44'></span><span id='citeF-46'></span>[[#cite-39|[39,41,44,46]]].
-==3 Definition of the characteristic length vector==
+==3. Definition of the characteristic length vector==
 The characteristic length vector <math display="inline">\boldsymbol h</math> is designed so that the expression for all the stabilization matrices and vectors reduce to those given for the 1D case as reported in <span id='citeF-61'></span>[[#cite-61|[61]]].

@@ Line 230: / Line 230: @@
 The characteristic length vectors in the r.h.s. of Eq.[[#eq-10|10]] are defined as follows.
-==='''Streamline characteristic length vector''', h<sub>v</sub>===
+'''''Streamline characteristic length vector''''', '''h'''<sub>v</sub>
 In Eq.[[#eq-10|10]] <math display="inline">{\boldsymbol h}_v</math> is a length vector along the velocity direction defined as
@@ Line 247: / Line 247: @@
 where <math display="inline">\alpha _v</math> is a streamline stabilization parameter, <math display="inline">l_v</math> is a characteristic element dimension and <math display="inline">\hat {\boldsymbol v}</math> is a unit velocity vector.
-==='''Absorption characteristic length vector''', h<sub>r</sub>===
+'''''Absorption characteristic length vector''''', '''h'''<sub>r</sub>
 In Eq.[[#eq-10|10]] <math display="inline">{\boldsymbol h}_r</math> is a characteristic length vector induced by absorption effects and defined as
@@ Line 296: / Line 296: @@
 ===<u>Any other element:</u> Dₛ=[0]===
-==='''Shock capturing characteristic length vector''' h<sub>s<sub>c</sub></sub>===
+'''''Shock capturing characteristic length vector''''' '''h'''<sub>s<sub>c</sub></sub>
 In Eq.[[#eq-10|10]] <math display="inline">{\boldsymbol h}_{sc}</math> is a shock-capturing characteristic length vector in the direction of the gradient of the solution. This vector accounts for the Gibbs oscillations across characteristic internal/boundary layers for advection-diffusion problems. It is defined as

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 In this work we extend to the multidimensional transient case the 1D FIC-FEM formulation presented in <span id='citeF-62'></span>[[#cite-62|[62]]]. The lay-out of the paper is the following. In the next section we formulate the FIC form of the equations governing multidimensional transient convection-diffusion-absorption problems. The finite element discretization is then presented. The stabilization parameters are obtained as an extension of the expression for the stationary case <span id='citeF-61'></span>[[#cite-61|[61]]]. The  accuracy of the multidimensional transient FIC-FEM formulation is verified in the solution of a number of 2D transient  advection-diffusion-absorption problems using uniform meshes of 3-noded triangles and 4-noded quadrilateral elements. Accurate solutions are obtained in all cases.
-==2 The multidimensional transient advection-diffusion-reaction problem==
+==2. The multidimensional transient advection-diffusion-reaction problem==
 ===2.1 Governing equations===
-==='''Transport balance'''===
+'''''Transport balance'''''
 The transport balance equation in a domain of area/volume <math display="inline">\Omega </math> can be expressed as
@@ Line 84: / Line 84: @@
 In Eqs.([[#eq-1.a|1.a]])&#8211;([[#eq-1.b|1.b]]) <math display="inline">\phi </math> is the transported variable (i.e., the temperature in a heat transfer problem or the concentration in a mass transfer problem), <math display="inline">v_i</math> is the <math display="inline">i</math>th component of the velocity vector <math display="inline">{\mathbf{}v}</math>; <math display="inline">\rho ,c</math> and <math display="inline">k_i</math> are the  density, the specific flux parameter and the conductivity of the material along the <math display="inline">i</math>th global direction, respectively and <math display="inline">s</math> is the reaction parameter. In the following, and unless otherwise specified, we will assume that the velocity field is solenoidal and that the problem parameters (<math display="inline">\rho ,  c ,k,s</math>) are constant over the analysis domain <math display="inline">\Omega </math>.
-==='''Boundary conditions'''===
+'''''Boundary conditions'''''
 <span id="eq-3"></span>
@@ Line 164: / Line 164: @@
 The governing equations [[#eq-1.a|1.a]] and [[#eq-1.b|1.b]] and the boundary conditions [[#eq-3|3]]-[[#eq-6|6]] are expressed using the FIC theory as <span id='citeF-55'></span>[[#cite-55|[55]]].
-===Transport balance===
+'''''Transport balance'''''
 <span id="eq-8"></span>
@@ Line 179: / Line 179: @@
 with <math display="inline">{\boldsymbol h}= [h_1,h_2,h_3]^T</math> in 3D.
-===Boundary conditions===
+'''''Boundary conditions'''''
 <span id="eq-9.a"></span>