In this paper, we present the construction and computational assessment of an hp-version discontinuous Galerkin finite element method (DGFEM) for the numerical solution of the Navier-Stokes equations governing 2D stationary incompressible flows. Using a stream-function formulation, which ensures that the incompressibility constraint is automatically satisfied, we reduce the system of Navier-Stokes equations to a single fourth-order nonlinear partial differential equation. We introduce a discretization of this fourth-order nonlinear partial differential equation based on a combination of the symmetric DGFEM for the biharmonic part of the equation and a DGFEM with jump-penalty terms for the hyperbolic part of the problem, and then we solve the resulting nonlinear problem using Newton's method. Numerical examples, including the solution of the 2D lid-driven cavity flow problem, are presented to demonstrate the convergence and accuracy of the method.
{"title":"Discontinuous Galerkin finite element approximation of the two-dimensional Navier-Stokes equations in stream-function formulation","authors":"I. Mozolevski, E. Süli, P. R. Bösing","doi":"10.1002/CNM.944","DOIUrl":"https://doi.org/10.1002/CNM.944","url":null,"abstract":"In this paper, we present the construction and computational assessment of an hp-version discontinuous Galerkin finite element method (DGFEM) for the numerical solution of the Navier-Stokes equations governing 2D stationary incompressible flows. Using a stream-function formulation, which ensures that the incompressibility constraint is automatically satisfied, we reduce the system of Navier-Stokes equations to a single fourth-order nonlinear partial differential equation. We introduce a discretization of this fourth-order nonlinear partial differential equation based on a combination of the symmetric DGFEM for the biharmonic part of the equation and a DGFEM with jump-penalty terms for the hyperbolic part of the problem, and then we solve the resulting nonlinear problem using Newton's method. Numerical examples, including the solution of the 2D lid-driven cavity flow problem, are presented to demonstrate the convergence and accuracy of the method.","PeriodicalId":51245,"journal":{"name":"Communications in Numerical Methods in Engineering","volume":"23 1","pages":"447-459"},"PeriodicalIF":0.0,"publicationDate":"2006-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/CNM.944","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"51602129","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}
A new variant of the FETI method for numerical solution of elliptic PDE is presented. The basic idea is to simplify inversion of the stiffness matrices of subdomains by using Lagrange multipliers not only for gluing the subdomains along the auxiliary interfaces, but also for implementation of the Dirichlet boundary conditions. Results of numerical experiments are presented which indicate that the new method may be even more efficient then the original FETI.
{"title":"Total FETI—an easier implementable variant of the FETI method for numerical solution of elliptic PDE","authors":"Z. Dostál, D. Hořák, R. Kučera","doi":"10.1002/CNM.881","DOIUrl":"https://doi.org/10.1002/CNM.881","url":null,"abstract":"A new variant of the FETI method for numerical solution of elliptic PDE is presented. The basic idea is to simplify inversion of the stiffness matrices of subdomains by using Lagrange multipliers not only for gluing the subdomains along the auxiliary interfaces, but also for implementation of the Dirichlet boundary conditions. Results of numerical experiments are presented which indicate that the new method may be even more efficient then the original FETI.","PeriodicalId":51245,"journal":{"name":"Communications in Numerical Methods in Engineering","volume":"22 1","pages":"1155-1162"},"PeriodicalIF":0.0,"publicationDate":"2006-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/CNM.881","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"51599350","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}
F. Rochinha, G. Alvarez, E. G. D. D. Carmo, A. Loula
In (Comput. Methods Appl. Mech. Eng. 2006, in press) we introduced a discontinuous Galerkin finite element method for Helmholtz equation in which continuity is relaxed locally in the interior of the element. The shape functions associated with interior nodes of the element are bilinear discontinuous bubbles, and the corresponding degrees of freedom can be eliminated at element level by static condensation yielding a global finite element method with the same connectivity of classical C° Galerkin finite element approximations. Stability is provided by the discontinuous bubbles with appropriate choice of the stabilization parameters related to the weak enforcement of continuity inside each element. In the present work, departing from the stencil obtained by condensation of the bubble degrees of freedom, we build a new strategy for determining the optimal values of these parameters aiming at matching the exact wave number in two different directions. Stability and accuracy of the proposed formulation are demonstrated in several numerical examples.
(第一版。方法:。动力机械。(Eng. 2006, in press)我们介绍了一种不连续的Galerkin有限元方法，其中连续性在单元内部局部松弛。与单元内部节点相关的形状函数是双线性不连续的气泡，通过静态凝聚可以在单元级消除相应的自由度，从而产生具有经典C°伽辽金有限元近似相同连性的全局有限元方法。稳定性是由不连续气泡提供的，稳定参数的适当选择与每个单元内部连续性的弱执行有关。在本工作中，我们从气泡自由度的凝结得到的模板出发，建立了一种新的策略来确定这些参数的最优值，目的是在两个不同方向上匹配精确的波数。数值算例验证了该公式的稳定性和准确性。
{"title":"A locally discontinuous enriched finite element formulation for acoustics","authors":"F. Rochinha, G. Alvarez, E. G. D. D. Carmo, A. Loula","doi":"10.1002/CNM.946","DOIUrl":"https://doi.org/10.1002/CNM.946","url":null,"abstract":"In (Comput. Methods Appl. Mech. Eng. 2006, in press) we introduced a discontinuous Galerkin finite element method for Helmholtz equation in which continuity is relaxed locally in the interior of the element. The shape functions associated with interior nodes of the element are bilinear discontinuous bubbles, and the corresponding degrees of freedom can be eliminated at element level by static condensation yielding a global finite element method with the same connectivity of classical C° Galerkin finite element approximations. Stability is provided by the discontinuous bubbles with appropriate choice of the stabilization parameters related to the weak enforcement of continuity inside each element. In the present work, departing from the stencil obtained by condensation of the bubble degrees of freedom, we build a new strategy for determining the optimal values of these parameters aiming at matching the exact wave number in two different directions. Stability and accuracy of the proposed formulation are demonstrated in several numerical examples.","PeriodicalId":51245,"journal":{"name":"Communications in Numerical Methods in Engineering","volume":"23 1","pages":"623-637"},"PeriodicalIF":0.0,"publicationDate":"2006-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/CNM.946","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"51602229","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}
M. V. Rodrigues, Fabrício Nogueira Correa, B. P. Jacob
This work presents the implementation of optimized numerical tools for the coupled analysis of floating platforms for offshore oil exploitation. The focus is on time-domain, nonlinear dynamic analysis, considering the coupling between the hydrodynamic behaviour of the hull and the structural behaviour of the mooring lines and risers modelled by finite elements (FEs). Some aspects of the formulation and solution of the large-amplitude equations of motion of the hull of the platform are presented, including a brief description of the hydrodynamic models and calculation of the environmental forces. The main aspects of the formulation for the spatial and time discretization of the structural model for the lines are also discussed. Since coupled analyses may require excessive computational costs, the objective of this work is to present the implementation and application of domain decomposition methods, adapted and specialized for the problem at hand, in order to optimize the efficiency of the computational tool. Two groups of domain decomposition methods are considered: the first is a subcycling technique that takes into account the natural partition that exists between the hull and the lines; the second considers the internal decomposition of the mesh of FEs to represent the mooring lines and risers. The methods are devised having in mind their implementation in computers with parallel architecture. Results of a numerical application are presented in order to assess the performance of the methods.
{"title":"Implicit domain decomposition methods for coupled analysis of offshore platforms","authors":"M. V. Rodrigues, Fabrício Nogueira Correa, B. P. Jacob","doi":"10.1002/CNM.945","DOIUrl":"https://doi.org/10.1002/CNM.945","url":null,"abstract":"This work presents the implementation of optimized numerical tools for the coupled analysis of floating platforms for offshore oil exploitation. The focus is on time-domain, nonlinear dynamic analysis, considering the coupling between the hydrodynamic behaviour of the hull and the structural behaviour of the mooring lines and risers modelled by finite elements (FEs). Some aspects of the formulation and solution of the large-amplitude equations of motion of the hull of the platform are presented, including a brief description of the hydrodynamic models and calculation of the environmental forces. The main aspects of the formulation for the spatial and time discretization of the structural model for the lines are also discussed. Since coupled analyses may require excessive computational costs, the objective of this work is to present the implementation and application of domain decomposition methods, adapted and specialized for the problem at hand, in order to optimize the efficiency of the computational tool. Two groups of domain decomposition methods are considered: the first is a subcycling technique that takes into account the natural partition that exists between the hull and the lines; the second considers the internal decomposition of the mesh of FEs to represent the mooring lines and risers. The methods are devised having in mind their implementation in computers with parallel architecture. Results of a numerical application are presented in order to assess the performance of the methods.","PeriodicalId":51245,"journal":{"name":"Communications in Numerical Methods in Engineering","volume":"23 1","pages":"599-621"},"PeriodicalIF":0.0,"publicationDate":"2006-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/CNM.945","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"51602193","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}
Injectivity loss occurs when sea water is injected in offshore fields or during reinjection of produced water that contains solid particles, oil drops and bacteria, each one with its own capability of reducing the permeability in the neighbourhood of the well. Knowledge regarding the causes of injectivity decline and its time scale is important for the prevention and remediation of this problem. We present a numerical model for water injection with retention of solid particles in a vertically stratified reservoir in a cylindrical axisymmetric geometry. The process of migration and retention of solid particles in porous media is studied through computer simulations. To do so, we apply a hydrodynamical formulation for the flow of water containing suspension particles. The numerical tools used in this work are the modified method of characteristics proposed by Douglas and Russell for the concentration of particles (suspended and retained) coupled with the mixed finite element method proposed by Raviart and Thomas for velocity and pressure. We briefly describe the method and show the numerical results. There is a strong tendency for the flow to remain confined in each layer.
{"title":"Numerical simulation of injectivity loss in stratified reservoirs","authors":"J. Wróbel, A. Nachbin, D. Marchesin","doi":"10.1002/CNM.911","DOIUrl":"https://doi.org/10.1002/CNM.911","url":null,"abstract":"Injectivity loss occurs when sea water is injected in offshore fields or during reinjection of produced water that contains solid particles, oil drops and bacteria, each one with its own capability of reducing the permeability in the neighbourhood of the well. Knowledge regarding the causes of injectivity decline and its time scale is important for the prevention and remediation of this problem. We present a numerical model for water injection with retention of solid particles in a vertically stratified reservoir in a cylindrical axisymmetric geometry. The process of migration and retention of solid particles in porous media is studied through computer simulations. To do so, we apply a hydrodynamical formulation for the flow of water containing suspension particles. The numerical tools used in this work are the modified method of characteristics proposed by Douglas and Russell for the concentration of particles (suspended and retained) coupled with the mixed finite element method proposed by Raviart and Thomas for velocity and pressure. We briefly describe the method and show the numerical results. There is a strong tendency for the flow to remain confined in each layer.","PeriodicalId":51245,"journal":{"name":"Communications in Numerical Methods in Engineering","volume":"53 1","pages":"507-520"},"PeriodicalIF":0.0,"publicationDate":"2006-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/CNM.911","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"51600629","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}
P. Capriles, L. D. Fonseca, H. Barbosa, Afonso C. C. Lemonge
One variant of the ant colony optimization (ACO) metaheuristic, known as rank-based ant system (RBAS), is proposed for weight minimization of structures involving discrete design variables. Stress and displacements constraints are handled using a penalty technique, and the structural analysis is performed by the finite element method. Results obtained for several discrete weight minimization problems show that the RBAS algorithm implemented is effective and competitive when compared with results found by genetic algorithms and another RBAS variant.
{"title":"Rank‐based ant colony algorithms for truss weight minimization with discrete variables","authors":"P. Capriles, L. D. Fonseca, H. Barbosa, Afonso C. C. Lemonge","doi":"10.1002/CNM.912","DOIUrl":"https://doi.org/10.1002/CNM.912","url":null,"abstract":"One variant of the ant colony optimization (ACO) metaheuristic, known as rank-based ant system (RBAS), is proposed for weight minimization of structures involving discrete design variables. Stress and displacements constraints are handled using a penalty technique, and the structural analysis is performed by the finite element method. Results obtained for several discrete weight minimization problems show that the RBAS algorithm implemented is effective and competitive when compared with results found by genetic algorithms and another RBAS variant.","PeriodicalId":51245,"journal":{"name":"Communications in Numerical Methods in Engineering","volume":"23 1","pages":"553-575"},"PeriodicalIF":0.0,"publicationDate":"2006-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/CNM.912","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"51600635","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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