Most existing Finite Element Method and the Material Point Method (FEM-MPM) coupling is designed for explicit solvers. By contrast, implicit schemes offer the advantage of substantially larger time steps while maintai...
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Most existing Finite Element Method and the Material Point Method (FEM-MPM) coupling is designed for explicit solvers. By contrast, implicit schemes offer the advantage of substantially larger time steps while maintaining enhanced stability, particularly beneficial for tackling stiff nonlinear problems. Despite this, the development of implicit FEM-MPM coupling has not been extensively explored, leaving a notable gap in the context of contact and elastoplastic deformation challenges. Thus, this paper proposes a novel unified FEM-MPM coupling approach within implicit time integration under the framework of multivariable variational principle and convex cone programming, termed CP-FEMP. The CP-FEMP is the first successful attempt to impose the contact constraints via Lagrange multiplier and barrier method under convex cone programming, which can tackle not only the tie constraints but also the frictional contact between MPM and FEM domains with ensuring convergence and feasibility regardless of the time step size or the mesh resolutions. The contact locking issue in tie contact is circumvented using a well-defined interpolation space. The governing equations, associated frictional contact model, and associated elastoplastic constitutive law are formulated into a global convex optimisation problem, which is efficiently solved using primal-dual interior-point method. Through a succession of standard contact and elastoplastic benchmarks, the CP-FEMP demonstrates its proficiency in the precise transference of contact forces across MPM and FEM domains while showcasing commendable energy conservation attributes. Finally, the CP-FEMP is applied to a slope-retaining wall interaction problem. All results demonstrate CP-FEMP provides a comprehensive solution for FEM-MPM coupling, allowing for large incremental step under nonlinear contact , elastoplastic large deformation and guaranteeing strict, hard non -penetration conditions without convergence issues.
For conventional Material Point Method (MPM), both explicit-based and implicit-based MPM have shortcomings: explicit MPM has high requirements on time steps, and implicit MPM has high requirements on convergence. To c...
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For conventional Material Point Method (MPM), both explicit-based and implicit-based MPM have shortcomings: explicit MPM has high requirements on time steps, and implicit MPM has high requirements on convergence. To circumvent these limitations, this paper innovatively proposes a convex cone programming-based implicit MPM (CP-MPM) algorithm, which ensures excellent convergence of solving complex problems involving large deformation, regardless of the chosen time step. In the proposed CP-MPM, the governing equations are initially transformed into a stationary point of a multivariable functional, leveraging the generalized Hellinger-Reissner (HR) variational principle. This stationary point problem is subsequently reformulated as a min-max convexcone optimization problem, with constraints originating from elastoplastic constitutive equations. In addition, a novel particle-based adhesive-frictional contact algorithm is proposed to effectively tackle the interaction between MPM domain and rigid bodies. The contact inequality between material points and rigid bodies is transformed into convexcone constraints, which rigorously prevent material point penetration and facilitate the imposition of irregular boundary conditions. Both elastic and elastoplastic problems involving contact under static or dynamic loading are ultimately represented as standard second-order coneprogramming (SOCP) problems, which is effectively solved by employing the Primal-Dual Interior Point (PDIP) method. The robustness, accuracy and convergence of the proposed method are validated through a series of elastic and elastoplastic benchmark problems. All results demonstrate the CP-MPM is a very promising method for implicitly solving complex practices.
In this paper we consider regularized convex cone programming problems. In particular, we first propose an iterative hard thresholding (IHT) method and its variant for solving regularized box constrained convex progra...
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In this paper we consider regularized convex cone programming problems. In particular, we first propose an iterative hard thresholding (IHT) method and its variant for solving regularized box constrained convexprogramming. We show that the sequence generated by these methods converges to a local minimizer. Also, we establish the iteration complexity of the IHT method for finding an -local-optimal solution. We then propose a method for solving regularized convex cone programming by applying the IHT method to its quadratic penalty relaxation and establish its iteration complexity for finding an -approximate local minimizer. Finally, we propose a variant of this method in which the associated penalty parameter is dynamically updated, and show that every accumulation point is a local izer of the problem.
We develop and apply a previously undescribed framework that is designed to extract information in the form of a positive definite kernel matrix from possibly crude, noisy, incomplete, inconsistent dissimilarity infor...
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We develop and apply a previously undescribed framework that is designed to extract information in the form of a positive definite kernel matrix from possibly crude, noisy, incomplete, inconsistent dissimilarity information between pairs of objects, obtainable in a variety of contexts. Any positive definite kernel defines a consistent set of distances, and the fitted kernel provides a set of coordinates in Euclidean space that attempts to respect the information available while controlling for complexity of the kernel. The resulting set of coordinates is highly appropriate for visualization and as input to classification and clustering algorithms. The framework is formulated in terms of a class of optimization problems that can be solved efficiently by using modern convex cone programming software. The power of the method is illustrated in the context of protein clustering based on primary sequence data. An application to the globin family of proteins resulted in a readily visualizable 3D sequence space of globins, where several subfamilies and subgroupings consistent with the literature were easily identifiable.
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