The patch test for mixed formulations

Zienkiewicz, O.C.; Qu, Shuang; Taylor, Robert L.; Nakazawa, S.

doi:10.1002/nme.1620231007

Cited by 195 publications

(88 citation statements)

References 9 publications

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“…Bilinear shape functions with chemical potential and displacements at the corner nodes are used to calculate the solution as a simplification. Réthoré et al [34] show that these approximations are sufficient for structural response in non-propagating cracks when the mesh is fine enough and do not produce issues related to the Babuska-Brezzi conditions [41]. This condition requires a higher order interpolation of the displacement field than of the pressure field.…”

Section: Discussionmentioning

confidence: 99%

Two-Dimensional Mode I Crack Propagation in Saturated Ionized Porous Media Using Partition of Unity Finite Elements

Kraaijeveld¹,

Huyghe

Remmers³

et al. 2013

Journal of Applied Mechanics

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Section: Discussionmentioning

confidence: 99%

Two-Dimensional Mode I Crack Propagation in Saturated Ionized Porous Media Using Partition of Unity Finite Elements

Kraaijeveld¹,

Huyghe

Remmers³

et al. 2013

Journal of Applied Mechanics

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“…Bu model ile suya doygun zemin sonlu elemanlar modelinde tanımlanabilmiş ve analizlerde aşırı boşluk suyu basıncı gelişiminin elde edilmesi mümkün olabilmiştir. İki fazlı model, boşluklu ortam teorisine [14,15] dayanmakta olup boşluk suyunun davranış üzerinde etkili olduğu geoteknik problemlerin tanımlanması açısından elverişli bir modeldir. İki fazlı modelde katı faz zemin danelerini, sıvı faz ise boşluk suyunu ifade etmektedir (Şekil 1-a).…”

Section: Yöntem (Method)unclassified

Açik Deni̇z Rüzgar Enerji̇si̇ Türbi̇nleri̇ Teki̇l Kazik Temelleri̇ni̇n Çevresi̇ndeki̇ Boşluk Suyu Basinci Geli̇şi̇mi̇

Taşan¹,

Savidis²,

Akdağ³

2014

Gazi Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi

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ÖZETAçık deniz rüzgar türbin tasarımında yaygın olarak kullanılan geniş çaplı tekil kazıklar ağır rüzgar ve dalga yüküne maruz kalırlar. Söz konusu temel sistemlerinin davranışının incelenmesine ve güvenlikli tasarım yöntemlerinin geliştirilmesine ihtiyaç bulunmaktadır. Bu amaçla, geniş çaplı tekil kazık ile tasarlanan türbin temel sisteminin davranışı sonlu elemanlar yöntemi ile incelenmiştir. Zemin elemanları için iki fazlı ve üç boyutlu u20p8 olarak isimlendirilen sonlu elemanlar geliştirilmiştir. Böylelikle, nümerik analizler boyunca çalışma kapsamında odaklanılmış olan, kazığın çevresindeki zeminde boşluk suyu basıncı gelişiminin elde edilmesi mümkün olabilmiştir. Zemin hipoplastik malzeme modeli ile modellenmiştir. Analiz sonuçları, açık deniz yük koşullarında kazık çevresinde önemli derecede aşırı boşluk suyu basıncı gelişiminin kazık başı deplasmanlarını oldukça etkilediğini göstermiştir. Literatürde yer alan mevcut yöntemlerin geniş çaplı kazığın deplasman tahmininde yetersiz kaldığı bulgulanmıştır. Sonrasında, kazık çapı-zemin geçirimliliğine dayanan parametrik analizler yapılmıştır. Analizler sonucu aşırı boşluk suyu basıncı birikim potansiyeline bağlı, temel tasarımında kullanılabilecek, karşılıklı etkileşim diyagramı sunulmuştur.Anahtar kelimeler: Açık deniz rüzgar enerjisi türbinleri, tekil kazık temel, tekrarlı yük, hipoplastik malzeme modeli, iki fazlı model, nümerik analiz, aşırı boşluk suyu basıncı PORE WATER PRESSURE DEVELOPMENT AROUND THE MONOPILE FOUNDATIONS OF OFFSHORE WIND ENERGY CONVERTERS ABSTRACTOffshore wind energy converters (OWECs) require specific foundations due to special loading conditions. Large diameter monopiles are widely proposed for OWECs. The behavior of large diameter monopiles under cyclic lateral loads was investigated in this study by means of numerical simulation with finite element method. Special focus is given to the pore water pressure accumulation around the monopile. For this purpose a fully coupled two-phase finite element (u20p8) is developed. A hypoplastic constitutive model for the saturated sand soil is used in the numerical analyses. The results revealed that, pile head displacement is strongly dependent on excess pore water pressure (EPWP) accumulation around the monopile. Subsequently, pile diameter-hydraulic conductivity interaction diagram was derived with a parametric study to estimate the pore water pressure accumulation potential. The proposed interaction diagram enables a preliminary design tool of monopiles for offshore wind energy converters in saturated sand soils.

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“…is enforced and k = 1, 2, 3 for 3D problems. In Equation (4) and in the following, summation convention for repeated indexes is assumed. Again, note that the underlined terms in Equation (4) can be viewed as non-local forms of the original differential equations.…”

Section: Basic Concepts Of the Finite Calculus (Fic) Methodsmentioning

confidence: 99%

“…The use of a mixed formulation or a selective integration technique eliminates the volumetric locking in many elements. These methods, however, fail in some elements such as linear triangles and tetrahedra, due to lack of satisfaction of the Babuska-Brezzi conditions [1][2][3] or alternatively the mixed patch test [1,4,5] not being passed.…”

Section: Introductionmentioning

confidence: 99%

Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

Oñate

Rojek

Taylor

et al. 2004

Numerical Meth Engineering

114

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SUMMARYMany finite elements exhibit the so-called 'volumetric locking' in the analysis of incompressible or quasi-incompressible problems. In this paper, a new approach is taken to overcome this undesirable effect. The starting point is a new setting of the governing differential equations using a finite calculus (FIC) formulation. The basis of the FIC method is the satisfaction of the standard equations for balance of momentum (equilibrium of forces) and mass conservation in a domain of finite size and retaining higher order terms in the Taylor expansions used to express the different terms of the differential equations over the balance domain. The modified differential equations contain additional terms which introduce the necessary stability in the equations to overcome the volumetric locking problem. The FIC approach has been successfully used for deriving stabilized finite element and meshless methods for a wide range of advective-diffusive and fluid flow problems. The same ideas are applied in this paper to derive a stabilized formulation for static and dynamic finite element analysis of incompressible solids using linear triangles and tetrahedra. Examples of application of the new stabilized formulation to linear static problems as well as to the semi-implicit and explicit 2D and 3D non-linear transient dynamic analysis of an impact problem and a bulk forming process are presented.

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The patch test for mixed formulations

Cited by 195 publications

References 9 publications

Two-Dimensional Mode I Crack Propagation in Saturated Ionized Porous Media Using Partition of Unity Finite Elements

Two-Dimensional Mode I Crack Propagation in Saturated Ionized Porous Media Using Partition of Unity Finite Elements

Açik Deni̇z Rüzgar Enerji̇si̇ Türbi̇nleri̇ Teki̇l Kazik Temelleri̇ni̇n Çevresi̇ndeki̇ Boşluk Suyu Basinci Geli̇şi̇mi̇

Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

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