iMechanica - convergence
https://www.imechanica.org/taxonomy/term/4134
enAn efficient convergence test for the fixed point method
https://www.imechanica.org/node/22840
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/12296">fixed point method</a></div><div class="field-item odd"><a href="/taxonomy/term/4134">convergence</a></div><div class="field-item even"><a href="/taxonomy/term/12297">function derivative</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>The fixed point method consists to find the solution of F(X)=X.</p>
<p>One can not get fixed with the convergence condition |F'(X)|<1 because if the function has an optimum then |F'(X)|=0 even if the solution is not yet reached.</p>
<p> </p>
<p>We introduce an efficient convergence test with the condition:</p>
<p>|Xn+1 - Xn| ≤ epsilon1 And |F(Xn+1)-Xn+1| ≤ epsilon2</p>
</div></div></div>Wed, 07 Nov 2018 18:10:14 +0000mohammedlamine22840 at https://www.imechanica.orghttps://www.imechanica.org/node/22840#commentshttps://www.imechanica.org/crss/node/22840Converged, but to the right solution??
https://www.imechanica.org/node/17509
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/4134">convergence</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Hello eveybody,</p>
<p>I am confused in using numerical schemes: What can convergence of a numerical scheme tell us and what not??? lets assume that a numerical scheme does converge to "A result" or "A number", but how could we be sure that the scheme has actually converged to the "RIGHT solution" and not a WRONG one??</p>
<p>Any input is appreciated!</p>
<p>Setareh</p>
</div></div></div>Tue, 18 Nov 2014 13:57:52 +0000setareh17509 at https://www.imechanica.orghttps://www.imechanica.org/node/17509#commentshttps://www.imechanica.org/crss/node/17509ABAQUS Reinforced Concrete Beam Model
https://www.imechanica.org/node/15893
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><span>Hi All,</span></p>
<p><span>I am trying to model a simply supported reinforced concrete beam with longitudinal reinforcement. I am using 8-node solid elements for the concrete, and 2-node truss elements for the longitudinal reinforcement bars. I have embedded the reinforcement bars in the concrete using an Embedded Region constraint. I am then applying a displacement at the midspan (deflection control). I am using the Concrete Damaged Plasticity model for the concrete. My model begins to run but will not converge. Some of the error messages that I get are:</span></p>
<p><span>The plasticity/creep/connector friction algorithm did not converge at 1 points</span></p>
<p><span>***WARNING: THE SYSTEM MATRIX HAS 1 NEGATIVE EIGENVALUES.</span></p>
<p><span>***NOTE: THE SOLUTION APPEARS TO BE DIVERGING. CONVERGENCE IS JUDGED UNLIKELY.</span></p>
<p><span>***NOTE: MATERIAL CALCULATIONS FAILED TO CONVERGE OR WERE NOT ATTEMPTED AT ONE </span><br /><span>OR MORE POINTS. CONVERGENCE IS JUDGED UNLIKELY.</span></p>
<p>
<span>Does anyone have any thoughts of what I am missing?</span></p>
<p><span>Thanks</span></p>
</div></div></div><div class="field field-name-taxonomy-forums field-type-taxonomy-term-reference field-label-above"><div class="field-label">Forums: </div><div class="field-items"><div class="field-item even"><a href="/forum/109">Ask iMechanica</a></div></div></div><div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/962">software</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Free Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/289">ABAQUS</a></div><div class="field-item odd"><a href="/taxonomy/term/4134">convergence</a></div><div class="field-item even"><a href="/taxonomy/term/4195">concrete damaged plasticity</a></div></div></div>Wed, 08 Jan 2014 17:06:11 +0000UWStructEng15893 at https://www.imechanica.orghttps://www.imechanica.org/node/15893#commentshttps://www.imechanica.org/crss/node/15893ABAQUS Convergence problem while using CREEP subroutine
https://www.imechanica.org/node/15816
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/962">software</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/289">ABAQUS</a></div><div class="field-item odd"><a href="/taxonomy/term/493">Constitutive modelling</a></div><div class="field-item even"><a href="/taxonomy/term/4134">convergence</a></div><div class="field-item odd"><a href="/taxonomy/term/9385">CREEEP</a></div></div></div>Tue, 17 Dec 2013 10:08:53 +0000ehsan.hosseini@empa.ch15816 at https://www.imechanica.orghttps://www.imechanica.org/node/15816#commentshttps://www.imechanica.org/crss/node/15816Journal Club Theme of August 2012: Mesh-Dependence in Cohesive Element Modeling
https://www.imechanica.org/node/12899
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/395">cohesive zone model</a></div><div class="field-item odd"><a href="/taxonomy/term/834">crack growth</a></div><div class="field-item even"><a href="/taxonomy/term/879">cohesive element</a></div><div class="field-item odd"><a href="/taxonomy/term/4134">convergence</a></div><div class="field-item even"><a href="/taxonomy/term/5999">mesh dependence</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
The classical cohesive zone theory of fracture finds its origins in the pioneering works by Dugdale, Barenblatt and Rice [1–3]. In their work, fracture is regarded as a progressive phenomenon in which separation takes place across a cohesive zone ahead of the crack tip and is resisted by cohesive tractions. Cohesive zone models are widely adopted by scientists and engineers perhaps due to their straightforward implementation within the traditional finite element framework. Some of the mainstream technologies proposed to introduce the cohesive theory of fracture into finite element analysis are the eXtended Finite Element Method (X-FEM) and cohesive elements.
</p>
<p>
Sukumar et al. [4] first utilized the X-FEM for modeling 3D crack growth by adding a discontinuous function and the asymptotic crack tip field to the finite elements. Subsequently, the method was extended to account for cohesive cracks [5]. While the X-FEM approach can deal with arbitrary crack paths, it becomes increasingly complicated for problems involving pervasive fracture and fragmentation.
</p>
<p>
On the other hand, the cohesive element approach consists on the insertion of cohesive finite elements along the edges or faces of the 2D or 3D mesh correspondingly [6-9]. Even though this approach is well suited for problems involving pre-defined crack directions, a number of known issues affect its accuracy when dealing with simulations including arbitrary crack paths, e.g., problems with the propagation of elastic stress waves (artificial compliance), spurious crack tip speed effects (lift-off), and mesh dependent effects (c.f. [10] for a complete review). However, the robustness of the method makes it one of the most common approaches for pervasive fracture and fragmentation analysis.
</p>
<p>
Artificial compliance and lift-off effects can be avoided by using an initially rigid cohesive law [8] or, more elegantly, a discontinuous Galerkin formulation with an activation criterion for cohesive elements [11,12]. However, the problem of mesh dependency, more precisely mesh-induced anisotropy and mesh-induced toughness, is an active area of research.
</p>
<p>
When a mesh is introduced to represent the continuum fracture problem within the cohesive element formulation, a constraint is introduced into the problem due to the inability of the mesh to represent the shape of an arbitrary crack. If we think of an arbitrary crack as a rectifiable path (2D) or surface (3D), the ability of a mesh to represent a straight line (2D) or plane (3D) as the mesh is refined is a necessary condition for the convergence of the cohesive element approach [14]. In 2-dimensional problems, the ability of a mesh to represent a straight segment is characterized by the path deviation ratio, defined as the ratio between the shortest path on the mesh edges connecting two nodes, and the Euclidian distance between them (see Fig. 1a). It is desirable, then, for the path deviation ratio to be independent on the segment direction (to avoid mesh-induced anisotropy) and to tend to one as the mesh size tends to zero (to avoid, in the limit, mesh-induced toughness). A mesh that satisfies these requirements is said to be isoperimetric.
</p>
<p>
[img_assist | nid=12896 | title=Figure 1 | desc=(a) 4k mesh, (b) 4k mesh with nodal perturbation, (c) K-Means mesh, and (d) Conjugate-Directions mesh. In all meshes, the path deviation ratio is defined as the ratio between the length of the red solid line and the blue dashed line. | link=none | align=center | width=640 | height=480]
</p>
<p>
The work of Radin and Sadun [13] shows that the pinwheel tiling of the plane has the isoperimetric property. Based on this result, Papoulia et al. [14] observed that crack paths obtained from pinwheel meshes are more stable as the meshes are refined compared to other types of meshes. It is worth noting, however, that pinwheel meshes are hard to generate and there is no known extension to the 3-dimensional case. In a recent work, Paulino et al. [15] analyzed the behavior of 4k meshes modified by nodal perturbation and edge swap operators (see Fig. 1b). Their results show that the expected value of the path deviation ratio (over all possible directions) is decreased for a given mesh size. It is worth noting, however, that even though the mesh induced-toughness is reduced in this way, the meshes under consideration still exhibit a considerable anisotropy [16].
</p>
<p>
Recently, K-Means meshes generated by the application of Delaunay’s triangulation to nodes resulting from clustering random points (see Fig. 1c), have been proposed as a way of alleviating mesh-induced anisotropy while keeping acceptable triangle quality [16]. For reasonable mesh sizes, K-Means meshes exhibit the same mean value of the path deviation ratio as 4k meshes with nodal perturbation while being perfectly isotropic. In the same article, another type of mesh termed Conjugate-Directions mesh is introduced. Conjugate-Directions meshes are generated by the application of barycentric subdivision to K-Means meshes as depicted in Fig. 1d. In this way, the barycentric subdivision adds new directions to the existing K-Means mesh which tend to be orthogonal to the original directions as the K-Means tends to be smoother (i.e., as the K-Means algorithm is applied over a larger number of random points). This can be interpreted as enriching the set of directions provided by the original mesh with new conjugate directions. Consequently, Conjugate-Directions meshes exhibit the same isotropy observed in K-Means meshes while producing a drastically reduced mean value of the path deviation ratio for identical mesh sizes. Figure 2 shows the polar plot of the path deviation ratio vs. mesh direction for 4k, K-Means and Conjugate-Directions meshes.
</p>
<p>
</p>
<p>
[img_assist | nid=12897 | title=Figure 2 | desc= Polar plot of the path deviation ratio (minus 1) as a function of the mesh direction for the meshes under consideration. |<br />
link=none | align=center | width=352 | height=264]
</p>
<p>
</p>
<p>
Moreover, preliminary results show that the convergence of K-Means meshes in the sense of the mean value of the path deviation ratio is similar to that of 4k meshes as the mesh size is reduced [17], see Fig 3a. At the same time, the standard deviation decreases at a fastest rate when compared to 4k meshes as shown in Fig. 3b. However, numerical evidence shows that the path deviation ratio tends to saturate around 1.04 for both meshes. On the other hand, the same numerical experiment shows no indication of saturation for Conjugate-Directions meshes in the studied range of mesh-sizes. In summary, K-Means meshes are isotropic thus not providing preferred crack propagation directions. In addition to being isotropic, Conjugate-Directions meshes exhibit a better convergence behavior in the sense of the path deviation ratio, making them good candidates for cohesive element analysis of crack propagation problems where the crack path is not known a priori.
</p>
<p>
</p>
<p>
[img_assist | nid=12898 | title=Figure 3 | desc=convergence of the path deviation ratio; (a) its mean value, and (b) its standard deviation. |<br />
link=none | align=center | width=640 | height=480]
</p>
<p>
</p>
<p>
To conclude, we should emphasize that even though convergence in the sense of the path deviation ratio is a necessary condition for convergence of the cohesive element formulation, when dealing with finite meshes an occasional misalignment of an edge in the cohesive crack might be enough to cause the simulated crack path to diverge from the physical one. Needless to say, the issue of crack path convergence in the cohesive element formulation is still an open problem. The hope of many of the previously mentioned research efforts is that arbitrary crack propagation can be achieved through mesh design (pre-processing).
</p>
<p>
</p>
<p>
<strong>References</strong></p>
<p>[1] Dugdale, D. S., “Yielding of steel sheets containing slits,” Journal of the Mechanics and Physics of Solids, Vol. 8, No. 2, 1960, pp. 100–104.<br />
[2] Barenblatt, G. I., “The Mathematical Theory of Equilibrium Cracks in Brittle Fracture,” Advances in Applied Mechanics, Vol. 7, 1962, pp. 55–129.<br />
[3] Rice, J. R., “Mathematical analysis in the mechanics of fracture,” Fracture: an advanced treatise, Vol. 2, 1968, pp. 191– 311.<br />
[4] Sukumar, N., Moes, N., Moran, B. and Belytschko, T., “Extended finite element method for three-dimensional crack modeling,” International Journal for Numerical Methods in Engineering, Vol. 48, 2000, pp. 1549-1570.<br />
[5] Moes, N. and Belytschko, T., “Extended ﬁnite element method for cohesive crack growth,” Engineering Fracture Mechanics, Vol. 69, 2002, pp. 813-833.<br />
[6] Xu, X. P. and Needleman, A., “Numerical simulations of fast crack growth in brittle solids,” Journal of the Mechanics and Physics of Solids, Vol. 42, 1994, pp. 1397–1397.<br />
[7] Xu, X. P. and Needleman, A., “Numerical simulations of dynamic crack growth along an interface,” International Journal of Fracture, Vol. 74, 1995, pp. 289–324.<br />
[8] Camacho, G. T. and Ortiz, M., “Computational modeling of impact damage in brittle materials,” International Journal of Solids and Structures, Vol. 33, 1996, pp. 2899–2938.<br />
[9] Ortiz, M. and Pandolfi, A., “Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis,” International Journal for Numerical Methods in Engineering, Vol. 44, No. 9, 1999, pp. 1267–1282.<br />
[10] Seagraves, A. and Radovitzky, R., Advances in cohesive zone modeling of dynamic fracture, Springer Verlag, 2009.<br />
[11] Noels, L. and Radovitzky, R., “An explicit discontinuous Galerkin method for non-linear solid dynamics: Formulation, parallel implementation and scalability properties,” International Journal for Numerical Methods in Engineering, Vol. 74, 2008, pp. 1393-1420.<br />
[12] Radovitzky, R., Seagraves, A., Tupek, M., and Noels, L., “A scalable 3D fracture and fragmentation algorithm based on a hybrid, discontinuous Galerkin, cohesive element method,” Computer Methods in Applied Mechanics and Engineering, Vol. 200, No. 1-4, 2011, pp. 326-344.<br />
[13] Radin, C. and Sadun, L., “The isoperimetric problem for pinwheel tilings,” Communications in Mathematical Physics, Vol. 177, No. 1, 1996, pp. 255–263.<br />
[14] Papoulia, K.D., Vavasis, S.A., and Ganguly, P., “Spatial convergence of crack nucleation using a cohesive finite-element model on a pinwheel-based mesh,” International Journal for Numerical Methods in Engineering, Vol. 67, No. 1, 2006, pp. 1-16.<br />
[15] Paulino, G. H., Park, K., Celes, W., and Espinha, R., “Adaptive dynamic cohesive fracture simulation using nodal perturbation and edge-swap operators,” International Journal for Numerical Methods in Engineering, Vol. 84, No. 11, 2010, pp. 1303–1343.<br />
[16] Rimoli, J. J., Rojas, J. J., and Khemani, F. N., “On the mesh dependency of cohesive zone models for crack propagation analysis,” in 53rd AIAA Structures, Structural Dynamics, and Materials and Conference, Honolulu, HI, 2012.<br />
[17] Rimoli, J. J. and Rojas, J. J., “Meshing strategies for the alleviation of mesh-induced effects in cohesive element models,” submitted. Preprint: <a href="http://arxiv.org/abs/1302.1161">http://arxiv.org/abs/1302.1161</a></p>
</div></div></div>Sat, 04 Aug 2012 04:33:59 +0000Julian J. Rimoli12899 at https://www.imechanica.orghttps://www.imechanica.org/node/12899#commentshttps://www.imechanica.org/crss/node/12899Reg. Convergence and Mesh Density
https://www.imechanica.org/node/6748
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi</p>
<p>I am using a FORTRAN FE code for 4 quad elements with a viscoelastic material and with contact analysis. The routine follows a Newton Raphson iteration scheme and which makes the residual zero for convergence. I am using direct sparse matrix solver from the FORTRAN math library.</p>
<p>When I use relatively lesser number of elements, about 1500 the solution converges and results are as expected. However, for the same analysis if I use more elements say about 4000, then the analysis starts to diverge midway. The residual starts to increase exponentially and the solution doesn't converge.
</p>
<p>
What could be the reason for this and how can I overcome it? Any suggestions would be helpful.
</p>
<p>
Thanks, <br />
Shriram
</p>
<p>
Research Assistant
</p>
<p>
University of Florida
</p>
<p>
P.S. I even use an automatic time stepping scheme, but it doesn't help.
</p>
</div></div></div><div class="field field-name-taxonomy-forums field-type-taxonomy-term-reference field-label-above"><div class="field-label">Forums: </div><div class="field-items"><div class="field-item even"><a href="/forum/109">Ask iMechanica</a></div></div></div><div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Free Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/4134">convergence</a></div><div class="field-item odd"><a href="/taxonomy/term/4362">Mesh density</a></div><div class="field-item even"><a href="/taxonomy/term/4363">residual</a></div><div class="field-item odd"><a href="/taxonomy/term/4364">contact analysis.</a></div></div></div>Wed, 09 Sep 2009 14:03:32 +0000shrimad6748 at https://www.imechanica.orghttps://www.imechanica.org/node/6748#commentshttps://www.imechanica.org/crss/node/6748Numerical phase modeling of BTO nanostructure using Landis' model with FEAP
https://www.imechanica.org/node/5991
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/785">Fortran</a></div><div class="field-item odd"><a href="/taxonomy/term/3543">ferroelectrics</a></div><div class="field-item even"><a href="/taxonomy/term/4131">Landau equation</a></div><div class="field-item odd"><a href="/taxonomy/term/4132">Landis</a></div><div class="field-item even"><a href="/taxonomy/term/4133">FEAP</a></div><div class="field-item odd"><a href="/taxonomy/term/4134">convergence</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p> The difference to the common model used is the f,g terms. The 3D FEM formulation has been derived and a new fortran program is written. The derivative matrix is symmetric. But the calculation result does not converge. I have tried every means to get a converging result. But it doesn't work, the residual norm always increases.</p>
</div></div></div>Wed, 08 Jul 2009 09:01:07 +0000aquis_mech5991 at https://www.imechanica.orghttps://www.imechanica.org/node/5991#commentshttps://www.imechanica.org/crss/node/5991