In this paper, we propose a semi-smooth Newton method and a primal-dual active set strategy to solve dynamical contact problems with friction. The conditions of contact with Coulomb’s friction can be formulated in the form of a fixed point problem related to a quasi-optimization one thanks to the semi-smooth Newton method. This method is based on the use of the primal-dual active set (PDAS) strategy. The main idea here is to find the correct subset $mathcal{A}$ of nodes that are in contact (active) opposed to those which are not in contact (inactive). For each case, the nonlinear boundary condition is replaced by a suitable linear one. Numerical experiments on both hyper-elastic problems and rigid granular materials are presented to show the efficiency of the proposed method.
{"title":"Unified primal-dual active set method for dynamic frictional contact problems","authors":"Abide, Stéphane, Barboteu, Mikaël, Cherkaoui, Soufiane, Dumont, Serge","doi":"10.1186/s13663-022-00729-4","DOIUrl":"https://doi.org/10.1186/s13663-022-00729-4","url":null,"abstract":"In this paper, we propose a semi-smooth Newton method and a primal-dual active set strategy to solve dynamical contact problems with friction. The conditions of contact with Coulomb’s friction can be formulated in the form of a fixed point problem related to a quasi-optimization one thanks to the semi-smooth Newton method. This method is based on the use of the primal-dual active set (PDAS) strategy. The main idea here is to find the correct subset $mathcal{A}$ of nodes that are in contact (active) opposed to those which are not in contact (inactive). For each case, the nonlinear boundary condition is replaced by a suitable linear one. Numerical experiments on both hyper-elastic problems and rigid granular materials are presented to show the efficiency of the proposed method.","PeriodicalId":12293,"journal":{"name":"Fixed Point Theory and Applications","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-27DOI: 10.1186/s13663-022-00726-7
Matei, Andaluzia
In the present paper we consider a class of generalized saddle-point problems described by means of the following variational system: $$begin{aligned} &a(u,v-u)+b(v-u,lambda )+j(v)-j(u)+J(u,v)-J(u,u)geq (f,v-u)_{X}, &b(u,mu -lambda )-psi (mu )+psi (lambda )leq 0, end{aligned}$$ ( $vin Ksubseteq X$ , $mu in Lambda subset Y$ ), where $(X,(cdot,cdot )_{X})$ and $(Y,(cdot,cdot )_{Y})$ are Hilbert spaces. We use a fixed-point argument and a saddle-point technique in order to prove the existence of at least one solution. Then, we obtain uniqueness and stability results. Subsequently, we pay special attention to the case when our problem can be seen as a perturbed problem by setting $psi (cdot )=epsilon bar{psi}(cdot )$ $(epsilon >0)$ . Then, we deliver a convergence result for $epsilon to 0$ , the case $psi equiv 0$ appearing like a limit case. The theory is illustrated by means of examples arising from contact mechanics, focusing on models with multicontact zones.
{"title":"On a class of generalized saddle-point problems arising from contact mechanics","authors":"Matei, Andaluzia","doi":"10.1186/s13663-022-00726-7","DOIUrl":"https://doi.org/10.1186/s13663-022-00726-7","url":null,"abstract":"In the present paper we consider a class of generalized saddle-point problems described by means of the following variational system: $$begin{aligned} &a(u,v-u)+b(v-u,lambda )+j(v)-j(u)+J(u,v)-J(u,u)geq (f,v-u)_{X}, &b(u,mu -lambda )-psi (mu )+psi (lambda )leq 0, end{aligned}$$ ( $vin Ksubseteq X$ , $mu in Lambda subset Y$ ), where $(X,(cdot,cdot )_{X})$ and $(Y,(cdot,cdot )_{Y})$ are Hilbert spaces. We use a fixed-point argument and a saddle-point technique in order to prove the existence of at least one solution. Then, we obtain uniqueness and stability results. Subsequently, we pay special attention to the case when our problem can be seen as a perturbed problem by setting $psi (cdot )=epsilon bar{psi}(cdot )$ $(epsilon >0)$ . Then, we deliver a convergence result for $epsilon to 0$ , the case $psi equiv 0$ appearing like a limit case. The theory is illustrated by means of examples arising from contact mechanics, focusing on models with multicontact zones.","PeriodicalId":12293,"journal":{"name":"Fixed Point Theory and Applications","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-06DOI: 10.1186/s13663-022-00725-8
Shillor, Meir, Kuttler, Kenneth L.
This work establishes the existence of a weak solution to a new model for the process of debonding of two elastic 2D-bars caused by humidity and vibrations. A version of the model was first presented in the PCM-CMM-2019 conference in Krakow, Poland, and was published in (Shillor in J. Theor. Appl. Mech. 58(2): 295–305 2020). The existence of a weak solution is proved by regularizing the problem and then setting it in an abstract form that allows the use of tools for pseudo-differential operators and a fixed point theorem. Questions of further analysis of the solutions, effective numerical methods and simulations, as well as possible controls, are unresolved, yet.
{"title":"Analysis of a debonding model of two elastic 2D-bars","authors":"Shillor, Meir, Kuttler, Kenneth L.","doi":"10.1186/s13663-022-00725-8","DOIUrl":"https://doi.org/10.1186/s13663-022-00725-8","url":null,"abstract":"This work establishes the existence of a weak solution to a new model for the process of debonding of two elastic 2D-bars caused by humidity and vibrations. A version of the model was first presented in the PCM-CMM-2019 conference in Krakow, Poland, and was published in (Shillor in J. Theor. Appl. Mech. 58(2): 295–305 2020). The existence of a weak solution is proved by regularizing the problem and then setting it in an abstract form that allows the use of tools for pseudo-differential operators and a fixed point theorem. Questions of further analysis of the solutions, effective numerical methods and simulations, as well as possible controls, are unresolved, yet.","PeriodicalId":12293,"journal":{"name":"Fixed Point Theory and Applications","volume":"79 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a nonlinear elasticity problem in a bounded domain, its boundary is decomposed in three parts: lower, upper, and lateral. The displacement of the substance, which is the unknown of the problem, is assumed to satisfy the homogeneous Dirichlet boundary conditions on the upper part, and not homogeneous one on the lateral part, while on the lower part, friction conditions are considered. In addition, the problem is governed by a particular constitutive law of elasticity system with a strongly nonlinear strain tensor. The functional framework leads to using Sobolev spaces with variable exponents. The formulation of the problem leads to a variational inequality, for which we prove the existence and uniqueness of the solution of the associated variational problem.
{"title":"On a nonlinear elasticity problem with friction and Sobolev spaces with variable exponents","authors":"Boukrouche, Mahdi, Merouani, Boubakeur, Zoubai, Fayrouz","doi":"10.1186/s13663-022-00724-9","DOIUrl":"https://doi.org/10.1186/s13663-022-00724-9","url":null,"abstract":"We consider a nonlinear elasticity problem in a bounded domain, its boundary is decomposed in three parts: lower, upper, and lateral. The displacement of the substance, which is the unknown of the problem, is assumed to satisfy the homogeneous Dirichlet boundary conditions on the upper part, and not homogeneous one on the lateral part, while on the lower part, friction conditions are considered. In addition, the problem is governed by a particular constitutive law of elasticity system with a strongly nonlinear strain tensor. The functional framework leads to using Sobolev spaces with variable exponents. The formulation of the problem leads to a variational inequality, for which we prove the existence and uniqueness of the solution of the associated variational problem.","PeriodicalId":12293,"journal":{"name":"Fixed Point Theory and Applications","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce the class of rectangular quasi b-metric spaces as a generalization of rectangular metric spaces, rectangular quasi-metric spaces, rectangular b-metric spaces, define generalized $(alpha ,psi ) $ -contraction mappings and study fixed point results for the maps introduced in the setting of rectangular quasi b-metric spaces. Our results extend and generalize related fixed point results in the literature, in particular, the works of Karapinar and Lakzian (J. Funct. Spaces 2014:914398, 2014), Alharbi et al. (J. Math. Anal. 9(3):47–60, 2018), and Khuangsatung et al. (Thai J. Math. 2020:89–101, 2020) from rectangular quasi metric space and rectangular b-metric space to rectangular quasi b-metric spaces. We also provide examples in support of our main findings. Furthermore, we applied one of our results to determine the existence of a solution to an integral equation.
{"title":"Fixed point theorems for generalized ((alpha ,psi ))-contraction mappings in rectangular quasi b-metric spaces","authors":"Abagaro, Bontu Nasir, Tola, Kidane Koyas, Mamud, Mustefa Abduletif","doi":"10.1186/s13663-022-00723-w","DOIUrl":"https://doi.org/10.1186/s13663-022-00723-w","url":null,"abstract":"In this paper, we introduce the class of rectangular quasi b-metric spaces as a generalization of rectangular metric spaces, rectangular quasi-metric spaces, rectangular b-metric spaces, define generalized $(alpha ,psi ) $ -contraction mappings and study fixed point results for the maps introduced in the setting of rectangular quasi b-metric spaces. Our results extend and generalize related fixed point results in the literature, in particular, the works of Karapinar and Lakzian (J. Funct. Spaces 2014:914398, 2014), Alharbi et al. (J. Math. Anal. 9(3):47–60, 2018), and Khuangsatung et al. (Thai J. Math. 2020:89–101, 2020) from rectangular quasi metric space and rectangular b-metric space to rectangular quasi b-metric spaces. We also provide examples in support of our main findings. Furthermore, we applied one of our results to determine the existence of a solution to an integral equation.","PeriodicalId":12293,"journal":{"name":"Fixed Point Theory and Applications","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-28Print Date: 2022-05-01DOI: 10.3399/bjgp22X719393
Giles Dawnay
{"title":"Television: <i>This Is Going to Hurt</i>.","authors":"Giles Dawnay","doi":"10.3399/bjgp22X719393","DOIUrl":"10.3399/bjgp22X719393","url":null,"abstract":"","PeriodicalId":12293,"journal":{"name":"Fixed Point Theory and Applications","volume":"2012 1","pages":"233"},"PeriodicalIF":5.3,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11189049/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87956031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-11DOI: 10.1186/s13663-022-00721-y
Krejčí, Pavel, Petrov, Adrien
A problem of motion of a piezoelectric actuator in contact with an elasto-plastic obstacle is reformulated as a PDE in one spatial dimension with hysteresis in the bulk and on the contact boundary. The model is shown to dissipate energy in agreement with the principles of thermodynamics. The main result includes existence, uniqueness, and continuous data dependence of solutions.
{"title":"A contact problem for a piezoelectric actuator on an elasto-plastic obstacle","authors":"Krejčí, Pavel, Petrov, Adrien","doi":"10.1186/s13663-022-00721-y","DOIUrl":"https://doi.org/10.1186/s13663-022-00721-y","url":null,"abstract":"A problem of motion of a piezoelectric actuator in contact with an elasto-plastic obstacle is reformulated as a PDE in one spatial dimension with hysteresis in the bulk and on the contact boundary. The model is shown to dissipate energy in agreement with the principles of thermodynamics. The main result includes existence, uniqueness, and continuous data dependence of solutions.","PeriodicalId":12293,"journal":{"name":"Fixed Point Theory and Applications","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-01DOI: 10.1186/s13663-022-00720-z
Wang, Fei, Reddy, B. Daya
As an extension of the finite element method, the virtual element method (VEM) can handle very general polygonal meshes, making it very suitable for non-matching meshes. In (Wriggers et al. in Comput. Mech. 58:1039–1050, 2016), the lowest-order virtual element method was applied to solve the contact problem of two elastic bodies on non-matching meshes. The numerical experiments showed the robustness and accuracy of the virtual element scheme. In this paper, we establish a priori error estimate of the virtual element method for the contact problem and prove that the lowest-order VEM achieves linear convergence order, which is optimal.
虚元法作为有限元法的一种扩展，可以处理非常一般的多边形网格，使其非常适用于非匹配网格。在《Wriggers et al. In Comput》中。机械学报(58:1039-1050,2016)，采用最低阶虚元法求解两个弹性体在不匹配网格上的接触问题。数值实验证明了该方法的鲁棒性和准确性。本文建立了接触问题虚元法的先验误差估计，并证明了最低阶VEM达到线性收敛阶，是最优的。
{"title":"A priori error analysis of virtual element method for contact problem","authors":"Wang, Fei, Reddy, B. Daya","doi":"10.1186/s13663-022-00720-z","DOIUrl":"https://doi.org/10.1186/s13663-022-00720-z","url":null,"abstract":"As an extension of the finite element method, the virtual element method (VEM) can handle very general polygonal meshes, making it very suitable for non-matching meshes. In (Wriggers et al. in Comput. Mech. 58:1039–1050, 2016), the lowest-order virtual element method was applied to solve the contact problem of two elastic bodies on non-matching meshes. The numerical experiments showed the robustness and accuracy of the virtual element scheme. In this paper, we establish a priori error estimate of the virtual element method for the contact problem and prove that the lowest-order VEM achieves linear convergence order, which is optimal.","PeriodicalId":12293,"journal":{"name":"Fixed Point Theory and Applications","volume":"184 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-14DOI: 10.1186/s13663-022-00719-6
Ali, Bashir, Adam, Aisha A.
In this paper, an inertial S-iteration iterative process for approximating a common fixed point of a finite family of quasi-Bregman nonexpansive mappings is introduced and studied in a reflexive Banach space. A strong convergence theorem is proved. Some applications of the theorem are presented. The results presented here improve, extend, and generalize some recent results in the literature.
{"title":"An inertial s-iteration process for a common fixed point of a family of quasi-Bregman nonexpansive mappings","authors":"Ali, Bashir, Adam, Aisha A.","doi":"10.1186/s13663-022-00719-6","DOIUrl":"https://doi.org/10.1186/s13663-022-00719-6","url":null,"abstract":"In this paper, an inertial S-iteration iterative process for approximating a common fixed point of a finite family of quasi-Bregman nonexpansive mappings is introduced and studied in a reflexive Banach space. A strong convergence theorem is proved. Some applications of the theorem are presented. The results presented here improve, extend, and generalize some recent results in the literature.","PeriodicalId":12293,"journal":{"name":"Fixed Point Theory and Applications","volume":"85 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-14DOI: 10.1186/s13663-022-00715-w
Nacry, Florent, Sofonea, Mircea
We consider an abstract inclusion in a real Hilbert space, governed by an almost history-dependent operator and a time-dependent multimapping with prox-regular values. We establish the unique solvability of the inclusion under appropriate assumptions on the data. The proof is based on the arguments of monotonicity, fixed point, and prox-regularity. We then use our result in order to deduce some direct consequences, including an existence and uniqueness result for a class of sweeping processes associated with prox-regular sets. Finally, we provide an example in a finite dimensional case inspired by a rheological model in solid mechanics.
{"title":"History-dependent operators and prox-regular sweeping processes","authors":"Nacry, Florent, Sofonea, Mircea","doi":"10.1186/s13663-022-00715-w","DOIUrl":"https://doi.org/10.1186/s13663-022-00715-w","url":null,"abstract":"We consider an abstract inclusion in a real Hilbert space, governed by an almost history-dependent operator and a time-dependent multimapping with prox-regular values. We establish the unique solvability of the inclusion under appropriate assumptions on the data. The proof is based on the arguments of monotonicity, fixed point, and prox-regularity. We then use our result in order to deduce some direct consequences, including an existence and uniqueness result for a class of sweeping processes associated with prox-regular sets. Finally, we provide an example in a finite dimensional case inspired by a rheological model in solid mechanics.","PeriodicalId":12293,"journal":{"name":"Fixed Point Theory and Applications","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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