A reliable incremental method of computing the limit load in deformation plasticity based on compliance: Continuous and discrete setting

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• 作者：Jaroslav Haslinger^a ; ^b ; ^{hasling@karlin.mff.cuni.cz" class="auth_mail" title="E-mail the corresponding author} ; Sergey Repin^c ; ^d ; ^{repin@pdmi.ras.ru" class="auth_mail" title="E-mail the corresponding author} ; ^{serepin@jyu.fi" class="auth_mail" title="E-mail the corresponding author} ; Stanislav Sysala^b ; ^{stanislav.sysala@ugn.cas.cz" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Variational problems with linear growth energy ; Incremental limit analysis ; Elastic-perfectly plastic problems ; Finite element approximation
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：303
• 期：Complete
• 页码：156-170
• 全文大小：881 K

文摘

The aim of this paper is to introduce an enhanced incremental procedure that can be used for the numerical evaluation and reliable estimation of the limit load. A conventional incremental method of limit analysis is based on parametrization of the respective variational formulation by the loading parameter class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300917&_mathId=si10.gif&_user=111111111&_pii=S0377042716300917&_rdoc=1&_issn=03770427&md5=150b02220a414bc56b64c4e4334ed2ae" title="Click to view the MathML source">ζ∈(0,ζ_lim)class="mathContainer hidden">class="mathCode">$\zeta \in (0,{\zeta }_{lim})$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300917&_mathId=si11.gif&_user=111111111&_pii=S0377042716300917&_rdoc=1&_issn=03770427&md5=60ea35878481aeecff14743486d7fbdc" title="Click to view the MathML source">ζ_limclass="mathContainer hidden">class="mathCode">${\zeta }_{lim}$ is generally unknown. The enhanced incremental procedure is operated in terms of an inverse mapping class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300917&_mathId=si12.gif&_user=111111111&_pii=S0377042716300917&_rdoc=1&_issn=03770427&md5=13eb0f913d299a77b9d0af0881d53a29" title="Click to view the MathML source">ψ:α↦ζclass="mathContainer hidden">class="mathCode">$\psi :\alpha \mapsto \zeta$ where the parameter class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300917&_mathId=si13.gif&_user=111111111&_pii=S0377042716300917&_rdoc=1&_issn=03770427&md5=a12bca1d2ac7a28a67d19fda90869b63" title="Click to view the MathML source">αclass="mathContainer hidden">class="mathCode">$\alpha$ belongs to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300917&_mathId=si14.gif&_user=111111111&_pii=S0377042716300917&_rdoc=1&_issn=03770427&md5=dc9f0b72d97d251bd5c2c265bd51e955" title="Click to view the MathML source">(0,+∞)class="mathContainer hidden">class="mathCode">$(0,+\infty )$ and its physical meaning is work of applied forces at the equilibrium state. The function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300917&_mathId=si2.gif&_user=111111111&_pii=S0377042716300917&_rdoc=1&_issn=03770427&md5=11bf4981095f6f4fcac8492d4930040c" title="Click to view the MathML source">ψclass="mathContainer hidden">class="mathCode">$\psi$ is continuous, nondecreasing and its values tend to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300917&_mathId=si11.gif&_user=111111111&_pii=S0377042716300917&_rdoc=1&_issn=03770427&md5=60ea35878481aeecff14743486d7fbdc" title="Click to view the MathML source">ζ_limclass="mathContainer hidden">class="mathCode">${\zeta }_{lim}$ as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300917&_mathId=si17.gif&_user=111111111&_pii=S0377042716300917&_rdoc=1&_issn=03770427&md5=ee70330ebe084719f3f716b5cecf68c7" title="Click to view the MathML source">α→+∞class="mathContainer hidden">class="mathCode">$\alpha \to +\infty$. Reduction of the problem to a finite element subspace associated with a mesh class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300917&_mathId=si18.gif&_user=111111111&_pii=S0377042716300917&_rdoc=1&_issn=03770427&md5=d635d94ff66cec2e1d3db8834f1bb514" title="Click to view the MathML source">T_hclass="mathContainer hidden">class="mathCode">${\mathcal{T}}_h$ generates the discrete limit parameter class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300917&_mathId=si19.gif&_user=111111111&_pii=S0377042716300917&_rdoc=1&_issn=03770427&md5=5477246e50a535ad4f86c1e7ff39b813" title="Click to view the MathML source">ζ_lim,hclass="mathContainer hidden">class="mathCode">${\zeta }_{lim,h}$ and the discrete counterpart class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300917&_mathId=si20.gif&_user=111111111&_pii=S0377042716300917&_rdoc=1&_issn=03770427&md5=32f724d244ef54265b77ee1fdd26b7eb" title="Click to view the MathML source">ψ_hclass="mathContainer hidden">class="mathCode">${\psi }_h$ to the function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300917&_mathId=si2.gif&_user=111111111&_pii=S0377042716300917&_rdoc=1&_issn=03770427&md5=11bf4981095f6f4fcac8492d4930040c" title="Click to view the MathML source">ψclass="mathContainer hidden">class="mathCode">$\psi$. We prove pointwise convergence class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300917&_mathId=si22.gif&_user=111111111&_pii=S0377042716300917&_rdoc=1&_issn=03770427&md5=1bb21c51647266aad823ca23c018df77" title="Click to view the MathML source">ψ_h→ψclass="mathContainer hidden">class="mathCode">${\psi }_h\to \psi$ and specify a class of yield functions for which class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300917&_mathId=si23.gif&_user=111111111&_pii=S0377042716300917&_rdoc=1&_issn=03770427&md5=099c0f810b832a3fddbde44c63f91b0c" title="Click to view the MathML source">ζ_lim,h→ζ_limclass="mathContainer hidden">class="mathCode">${\zeta }_{lim,h}\to {\zeta }_{lim}$. These convergence results enable to find reliable lower and upper bounds of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300917&_mathId=si11.gif&_user=111111111&_pii=S0377042716300917&_rdoc=1&_issn=03770427&md5=60ea35878481aeecff14743486d7fbdc" title="Click to view the MathML source">ζ_limclass="mathContainer hidden">class="mathCode">${\zeta }_{lim}$. Numerical tests confirm computational efficiency of the suggested method.