Recovery of equilibrium on star patches from conforming finite elements with a linear basis

Maunder, E. A. W.; Almeida, J. P. Moitinho de

doi:10.1002/nme.3297

Cited by 7 publications

(20 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…After substituting from Equation , the fictitious term becomes the following:

- \int_{Ω_{i}^{e}} - (({bold-italicD}_{1}^{T} {normalΨ}_{i}^{e}) q_{c}^{e}) {boldr}_{i} + ({bold-italicD}_{2}^{T} {normalΨ}_{i}^{e}) m_{c}^{e} d Ω .

Now, in general, unlike in the case of the 2D problem discussed in the authors' earlier work,

\sum_{i} {bold-italicM}_{false[i false]}^{e} \neq bold0

. However, the terms in

{bold-italicM}_{false[i false]}^{e}

can be regrouped as follows:

\begin{matrix} alignleft \\ align-1 & align-2 {bold-italicM}_{[i]}^{e} = (\int_{{normalΩ}_{i}^{e}} {normalΨ}_{i}^{e} \bar{bold-italicc} d Ω + \int_{{normalΓ}_{i}^{e}} {normalΨ}_{i}^{e} \bar{bold-italicm} d Γ - \int_{{normalΩ}_{i}^{e}} ({bold-italicD}_{2}^{T} {normalΨ}_{i}^{e}) {bold-italicm}_{c}^{e} d Ω) \\ align-1 & align-2 + (- \int_{{normalΩ}_{i}^{e}} {normalΨ}_{i}^{e} \bar{p} {bold-italicr}_{i} d Ω - \int_{{normalΓ}_{i}^{e}} {normalΨ}_{i}^{e} q...) \end{matrix}

…”

Section: Recovery Of Equilibrium From Conforming Elementsmentioning

confidence: 99%

“…This means that the weighted applied transverse loads plus the fictitious pressures on a star patch are balanced in the translational sense but the total weighted applied and fictitious loads are not, in general, balanced in the rotational sense, ie, there is a resultant moment M [ i ] , and this is independent of where moments are taken about. Following the reasoning and notation in the authors' earlier work for 2D continuum problems, we can express the contribution to M [ i ] from the loads on element e as the sum of moments taken about the vertex i as follows:

M_{[i]}^{e} = {\bar{bold-italicM}}_{[i]}^{e} + M_{h [i]}^{e},

where the first contribution is derived from the weighted applied loads and the second from the fictitious loads. The complete expression for

{bold-italicM}_{false[i false]}^{e}

, with both terms as detailed within the parentheses, becomes

{bold-italicM}_{false[i false]}^{e} = ((), \int_{{normalΩ}_{i}^{e}} {normalΨ}_{i}^{e} ((), - {\overset{p}{true}}^{e} r_{i} + {\overset{c}{true}}^{e}) d normalΩ + \int_{{normalΓ}_{i}^{e}} {normalΨ}_{i}^{e} ((), - {\overset{q}{true}}_{n}^{e} r_{i} + {\overset{m}{true}}^{e}) d normalΓ) + ((), \int_{{normalΩ}_{i}^{e}} ((), - p_{i}^{e} r_{i} + c_{i}^{e}) d normalΩ),

where

r_{i} = ((), bold-italicx -)

…”

Section: Recovery Of Equilibrium From Conforming Elementsmentioning

confidence: 99%

See 1 more Smart Citation

Recovery of strong equilibrium from displacement‐based finite element models of Reissner‐Mindlin plates

Maunder

Almeida

2018

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

Summary In this paper, we extend to Reissner‐Mindlin plate bending problems a technique, originally proposed in the context of two‐dimensional and three‐dimensional continua, for recovering fully equilibrated stresses from the solution of a compatible finite element model. The technique involves partition of unity functions and the analyses of overlapping star patches modelled with hybrid equilibrium plate elements. The patches are subjected to balanced systems of loads composed of partitioned and fictitious loads, where the latter are derived from the stresses of the compatible solution. The special case of assumed linear displacement fields of both deflection and rotation for the compatible model is included. This case requires additional fields of stress resultants to correct possible rotational imbalances of star patches, and these are derived elementwise. Other cases of nonconforming elements are briefly considered. Numerical examples are presented to illustrate the effectiveness of these techniques in terms of the deviation of the recovery, which compares the complementary strain energy of a recovered solution with that obtained by a global equilibrated analysis based on the same stress approximations.

show abstract

“…After substituting from Equation , the fictitious term becomes the following:

- \int_{Ω_{i}^{e}} - (({bold-italicD}_{1}^{T} {normalΨ}_{i}^{e}) q_{c}^{e}) {boldr}_{i} + ({bold-italicD}_{2}^{T} {normalΨ}_{i}^{e}) m_{c}^{e} d Ω .

Now, in general, unlike in the case of the 2D problem discussed in the authors' earlier work,

\sum_{i} {bold-italicM}_{false[i false]}^{e} \neq bold0

. However, the terms in

{bold-italicM}_{false[i false]}^{e}

can be regrouped as follows:

\begin{matrix} alignleft \\ align-1 & align-2 {bold-italicM}_{[i]}^{e} = (\int_{{normalΩ}_{i}^{e}} {normalΨ}_{i}^{e} \bar{bold-italicc} d Ω + \int_{{normalΓ}_{i}^{e}} {normalΨ}_{i}^{e} \bar{bold-italicm} d Γ - \int_{{normalΩ}_{i}^{e}} ({bold-italicD}_{2}^{T} {normalΨ}_{i}^{e}) {bold-italicm}_{c}^{e} d Ω) \\ align-1 & align-2 + (- \int_{{normalΩ}_{i}^{e}} {normalΨ}_{i}^{e} \bar{p} {bold-italicr}_{i} d Ω - \int_{{normalΓ}_{i}^{e}} {normalΨ}_{i}^{e} q...) \end{matrix}

…”

Section: Recovery Of Equilibrium From Conforming Elementsmentioning

confidence: 99%

M_{[i]}^{e} = {\bar{bold-italicM}}_{[i]}^{e} + M_{h [i]}^{e},

where the first contribution is derived from the weighted applied loads and the second from the fictitious loads. The complete expression for

{bold-italicM}_{false[i false]}^{e}

, with both terms as detailed within the parentheses, becomes

{bold-italicM}_{false[i false]}^{e} = ((), \int_{{normalΩ}_{i}^{e}} {normalΨ}_{i}^{e} ((), - {\overset{p}{true}}^{e} r_{i} + {\overset{c}{true}}^{e}) d normalΩ + \int_{{normalΓ}_{i}^{e}} {normalΨ}_{i}^{e} ((), - {\overset{q}{true}}_{n}^{e} r_{i} + {\overset{m}{true}}^{e}) d normalΓ) + ((), \int_{{normalΩ}_{i}^{e}} ((), - p_{i}^{e} r_{i} + c_{i}^{e}) d normalΩ),

where

r_{i} = ((), bold-italicx -)

…”

Section: Recovery Of Equilibrium From Conforming Elementsmentioning

confidence: 99%

Recovery of strong equilibrium from displacement‐based finite element models of Reissner‐Mindlin plates

Maunder

Almeida

2018

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

“…Thus, we need to analyze the minimum degree required for the stress field to guarantee that the system of equations (5.14) is solvable. This procedure cannot be directly applied to bilinear FE elements since they do not guarantee the rotational equilibrium of the patch [123]. We will then consider bi-quadratic elements (Q8), therefore the FE stress field, σ h , will have quadratic terms.…”

Section: Comments About the Resolution Of The System Of Equationsmentioning

confidence: 99%

“…Note that, because of the lack of rotational equilibrium for the linear (Q4) FE solution [123], this method can be directly applied only for quadratic elements (Q8), as in the FER. For Q4 elements, it requires a post-processing of the FE solution in order to correct the lack of rotational equilibrium.…”

Section: The Recovery Procedures For the Auxiliary Stress Fieldmentioning

confidence: 99%

See 1 more Smart Citation

Cartesian grid FEM (cgFEM): High performance h-adaptive FE analysis with efficient error control. Application to structural shape optimization

Soriano¹

View full text Add to dashboard Cite

Recovery of Complementary Solutions

2016

Equilibrium Finite Element Formulations

View full text Add to dashboard Cite

Recovery of equilibrium on star patches from conforming finite elements with a linear basis

Cited by 7 publications

References 18 publications

Recovery of strong equilibrium from displacement‐based finite element models of Reissner‐Mindlin plates

Recovery of strong equilibrium from displacement‐based finite element models of Reissner‐Mindlin plates

Cartesian grid FEM (cgFEM): High performance h-adaptive FE analysis with efficient error control. Application to structural shape optimization

Recovery of Complementary Solutions

Contact Info

Product

Resources

About