An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions

This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions.To this end,wefirst reformulate the original problem into a minimax problem corresponding to a feasible augmented La-grangian,which can be solved by the augmented Lagrangian method in an infinite dimensional setting.Based on this,by expressing the primal and dual variables with two individual deep neural network functions,we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimiza-tion method together with a projection technique.Compared to the traditional penalty method,the new method admits two main advantages:i)the choice of the penalty parameter isflexible and robust,and ii)the numerical solution is more accurate in the same magnitude of computational cost.As typical applications,we apply the new ap-proach to solve elliptic problems and(nonlinear)eigenvalue problems with essential boundary conditions,and numerical experiments are presented to show the effective-ness of the new method.

A simple and robust method for essential boundary condition treatments in mesh-less methods

To improve the treatment efficiency of essential boundary condition in mesh-less methods, a simple and robust method is proposed in this paper. Rising weight of nodes in the construction of trail function, specified for essential boundary condition, can make the trail function pass through these nodes. And then, the trail function can satisfy the essential boundary condition previously by setting diagonal element to 1 or multiplying diagonal element by a big number in FEM. The MLS method is adopted to validate this method, and it is proved that this method is eostless and robust in most of mesh-less methods.

An efficient SPH methodology for modelling mechanical characteristics of particulate composites

Particulate composites are one of the widely used materials in producing numerous state-of-the-art components in biomedical,automobile,aerospace including defence technology.Variety of modelling techniques have been adopted in the past to model mechanical behaviour of particulate composites.Due to their favourable properties,particle-based methods provide a convenient platform to model failure or fracture of these composites.Smooth particle hydrodynamics(SPH)is one of such methods which demonstrate excellent potential for modelling failure or fracture of particulate composites in a Lagrangian setting.One of the major challenges in using SPH method for modelling composite materials depends on accurate and efficient way to treat interface and boundary conditions.In this paper,a masterslave method based multi-freedom constraints is proposed to impose essential boundary conditions and interfacial displacement constraints in modelling mechanical behaviour of composite materials using SPH method.The proposed methodology enforces the above constraints more accurately and requires only smaller condition number for system stiffness matrix than the procedures based on typical penalty function approach.A minimum cut-off value-based error criteria is employed to improve the computational efficiency of the proposed methodology.In addition,the proposed method is further enhanced by adopting a modified numerical interpolation scheme along the boundary to increase the accuracy and computational efficiency.The numerical examples demonstrate that the proposed master-slave approach yields better accuracy in enforcing displacement constraints and requires approximately the same computational time as that of penalty method.

Scaled boundary isogeometric analysis for 2D elastostatics

A new numerical method,scaled boundary isogeometric analysis(SBIGA)combining the concept of the scaled boundary finite element method(SBFEM)and the isogeometric analysis(IGA),is proposed in this study for 2D elastostatic problems with both homogenous and inhomogeneous essential boundary conditions.Scaled boundary isogeometric transformation is established at a specified scaling center with boundary isogeometric representation identical to the design model imported from CAD system,which can be automatically refined without communication with the original system and keeping geometry invariability.The field variable,that is,displacement,is constructed by the same basis as boundary isogeometric description keeping analytical features in radial direction.A Lagrange multiplier scheme is suggested to impose the inhomogeneous essential boundary conditions.The new proposed method holds the semi-analytical feature inherited from SBFEM,that is,discretization only on boundaries rather than the entire domain,and isogeometric boundary geometry from IGA,which further increases the accuracy of the solution.Numerical examples,including circular cavity in full plane,Timoshenko beam with inhomogenous boundary conditions and infinite plate with circular hole subjected to remotely tension,demonstrate that SBIGA can be applied efficiently to elastostatic problems with various boundary conditions,and powerful in accuracy of solution and less degrees of freedom(DOF)can be achieved in SBIGA than other methods.