The exact minimax penalty function method is used to solve a noncon- vex differentiable optimization problem with both inequality and equality constraints. The conditions for exactness of the penalization for the exac...The exact minimax penalty function method is used to solve a noncon- vex differentiable optimization problem with both inequality and equality constraints. The conditions for exactness of the penalization for the exact minimax penalty function method are established by assuming that the functions constituting the considered con- strained optimization problem are invex with respect to the same function η （with the exception of those equality constraints for which the associated Lagrange multipliers are negative these functions should be assumed to be incave with respect to η）. Thus, a threshold of the penalty parameter is given such that, for all penalty parameters exceeding this threshold, equivalence holds between the set of optimal solutions in the considered constrained optimization problem and the set of minimizer in its associated penalized problem with an exact minimax penalty function. It is shown that coercivity is not suf- ficient to prove the results.展开更多
We give an existence result of the obstacle parabolic equations3b（x,u） div（a（x,t,u, Vu））＋div（（x,t,u））=f in QT, 3twhere b（x,u） is bounded function ot u, the term atva,x,r,u, v u）） is a Letay type operat...We give an existence result of the obstacle parabolic equations3b（x,u） div（a（x,t,u, Vu））＋div（（x,t,u））=f in QT, 3twhere b（x,u） is bounded function ot u, the term atva,x,r,u, v u）） is a Letay type operator and the function is a nonlinear lower order and satisfy only the growth condition. The second term belongs to L1 （QT）. The proof of an existence solution is based on the penalization methods.展开更多
The existence condition of the solution of special nonlinear penalized equation of the linear complementarity problems is obtained by the relationship between penalized equations and an absolute value equation. Newton...The existence condition of the solution of special nonlinear penalized equation of the linear complementarity problems is obtained by the relationship between penalized equations and an absolute value equation. Newton method is used to solve penalized equation, and then the solution of the linear complementarity problems is obtained. We show that the proposed method is globally and superlinearly convergent when the matrix of complementarity problems of its singular values exceeds 0;numerical results show that our proposed method is very effective and efficient.展开更多
For the purpose of achieving high-resolution optimal solutions this paper proposes a nodal design variablebased adaptive method for topology optimization of continuum structures. The analysis mesh-independent density ...For the purpose of achieving high-resolution optimal solutions this paper proposes a nodal design variablebased adaptive method for topology optimization of continuum structures. The analysis mesh-independent density field, interpolated by the nodal design variables at a given set of density points, is adaptively refined/coarsened accord- ing to a criterion regarding the gray-scale measure of local regions. New density points are added into the gray regions and redundant ones are removed from the regions occupied by purely solid/void phases for decreasing the number of de- sign variables. A penalization factor adaptivity technique is employed-to prevent premature convergence of the optimiza- tion iterations. Such an adaptive scheme not only improves the structural boundary description quality, but also allows for sufficient further topological evolution of the structural layout in higher adaptivity levels and thus essentially enables high-resolution solutions. Moreover, compared with the case with uniformly and finely distributed density points, the proposed adaptive method can achieve a higher numerical efficiency of the optimization process.展开更多
The penalty equation of LCP is transformed into the absolute value equation, and then the existence of solutions for the penalty equation is proved by the regularity of the interval matrix. We propose a generalized Ne...The penalty equation of LCP is transformed into the absolute value equation, and then the existence of solutions for the penalty equation is proved by the regularity of the interval matrix. We propose a generalized Newton method for solving the linear complementarity problem with the regular interval matrix based on the nonlinear penalized equation. Further, we prove that this method is convergent. Numerical experiments are presented to show that the generalized Newton method is effective.展开更多
板条状结构的设计域具有较大的长宽比,常规的拓扑优化方法无法获得清晰的、周期性的拓扑形式或求解困难。以结构的最小柔度为目标函数,单元相对密度为设计变量,构建周期性拓扑优化问题的数学模型。提出一种基于变密度理论固体各向同性...板条状结构的设计域具有较大的长宽比,常规的拓扑优化方法无法获得清晰的、周期性的拓扑形式或求解困难。以结构的最小柔度为目标函数,单元相对密度为设计变量,构建周期性拓扑优化问题的数学模型。提出一种基于变密度理论固体各向同性微结构材料惩罚模型法(Solid isotropic microstructures with penalization,SIMP)的周期性拓扑优化的方法。在数学模型中设置额外的约束条件,保证优化结构可以得到周期性的拓扑形式。通过优化准则法推导出虚拟子域设计变量的迭代公式,利用体积约束计算出拉格朗日乘子。引入过滤函数解决拓扑优化容易出现数值计算不稳定,导致棋盘格、网格依赖性等问题。利用所提出的方法,通过平面矩形悬臂梁结构算例,获得平面矩形悬臂梁结构的周期性拓扑形式。结果表明,在优化过程中,出现周期性的孔洞。随着迭代次数的增加,孔洞数目没有增加,说明该方法具有较强的稳健性。子域数目取值不同时,均可以得到具有周期性的拓扑形式,且具有良好的一致性。展开更多
文摘The exact minimax penalty function method is used to solve a noncon- vex differentiable optimization problem with both inequality and equality constraints. The conditions for exactness of the penalization for the exact minimax penalty function method are established by assuming that the functions constituting the considered con- strained optimization problem are invex with respect to the same function η （with the exception of those equality constraints for which the associated Lagrange multipliers are negative these functions should be assumed to be incave with respect to η）. Thus, a threshold of the penalty parameter is given such that, for all penalty parameters exceeding this threshold, equivalence holds between the set of optimal solutions in the considered constrained optimization problem and the set of minimizer in its associated penalized problem with an exact minimax penalty function. It is shown that coercivity is not suf- ficient to prove the results.
文摘We give an existence result of the obstacle parabolic equations3b（x,u） div（a（x,t,u, Vu））＋div（（x,t,u））=f in QT, 3twhere b（x,u） is bounded function ot u, the term atva,x,r,u, v u）） is a Letay type operator and the function is a nonlinear lower order and satisfy only the growth condition. The second term belongs to L1 （QT）. The proof of an existence solution is based on the penalization methods.
文摘The existence condition of the solution of special nonlinear penalized equation of the linear complementarity problems is obtained by the relationship between penalized equations and an absolute value equation. Newton method is used to solve penalized equation, and then the solution of the linear complementarity problems is obtained. We show that the proposed method is globally and superlinearly convergent when the matrix of complementarity problems of its singular values exceeds 0;numerical results show that our proposed method is very effective and efficient.
基金supported by the Key Project of Chinese National Programs for Fundamental Research and Development(2010CB832703)the National Natural Science Foundation of China(11072047 and 91130025)
文摘For the purpose of achieving high-resolution optimal solutions this paper proposes a nodal design variablebased adaptive method for topology optimization of continuum structures. The analysis mesh-independent density field, interpolated by the nodal design variables at a given set of density points, is adaptively refined/coarsened accord- ing to a criterion regarding the gray-scale measure of local regions. New density points are added into the gray regions and redundant ones are removed from the regions occupied by purely solid/void phases for decreasing the number of de- sign variables. A penalization factor adaptivity technique is employed-to prevent premature convergence of the optimiza- tion iterations. Such an adaptive scheme not only improves the structural boundary description quality, but also allows for sufficient further topological evolution of the structural layout in higher adaptivity levels and thus essentially enables high-resolution solutions. Moreover, compared with the case with uniformly and finely distributed density points, the proposed adaptive method can achieve a higher numerical efficiency of the optimization process.
文摘The penalty equation of LCP is transformed into the absolute value equation, and then the existence of solutions for the penalty equation is proved by the regularity of the interval matrix. We propose a generalized Newton method for solving the linear complementarity problem with the regular interval matrix based on the nonlinear penalized equation. Further, we prove that this method is convergent. Numerical experiments are presented to show that the generalized Newton method is effective.
文摘板条状结构的设计域具有较大的长宽比,常规的拓扑优化方法无法获得清晰的、周期性的拓扑形式或求解困难。以结构的最小柔度为目标函数,单元相对密度为设计变量,构建周期性拓扑优化问题的数学模型。提出一种基于变密度理论固体各向同性微结构材料惩罚模型法(Solid isotropic microstructures with penalization,SIMP)的周期性拓扑优化的方法。在数学模型中设置额外的约束条件,保证优化结构可以得到周期性的拓扑形式。通过优化准则法推导出虚拟子域设计变量的迭代公式,利用体积约束计算出拉格朗日乘子。引入过滤函数解决拓扑优化容易出现数值计算不稳定,导致棋盘格、网格依赖性等问题。利用所提出的方法,通过平面矩形悬臂梁结构算例,获得平面矩形悬臂梁结构的周期性拓扑形式。结果表明,在优化过程中,出现周期性的孔洞。随着迭代次数的增加,孔洞数目没有增加,说明该方法具有较强的稳健性。子域数目取值不同时,均可以得到具有周期性的拓扑形式,且具有良好的一致性。
