Pavel Solin

Adaptive hp-FEM with dynamical meshes for transient heat and moisture transfer problems

We are concerned with the time-dependent multiphysics problem of heat and moisture transfer in the context of civil engineering applications. The problem is challenging due to its multiscale nature (temperature usually propagates orders of magnitude faster than moisture), different characters of the two fields (moisture exhibits boundary layers which are not present in the temperature field), extremely long integration times (30 years or more), and lack of viable error control mechanisms. In order to solve the problem efficiently, we employ a novel multimesh adaptive higher-order finite element method (hp-FEM) based on dynamical meshes and adaptive time step control. We investigate the possibility to approximate the temperature and humidity fields on individual dynamical meshes equipped with mutually independent adaptivity mechanisms. Numerical examples related to a realistic nuclear reactor vessel simulation are presented.

Monolithic discretization of linear thermoelasticity problems via adaptive multimesh hp-FEM

In linear thermoelasticity models, the temperature T and the displacement components u1,u2 exhibit large qualitative differences: while T typically is very smooth everywhere in the domain, the displacements u1,u2 have singular gradients (stresses) at re-entrant corners and edges. The mesh must be extremely fine in these areas so that stress intensity factors are resolved sufficiently. One of the best available methods for this task is the exponentially-convergent hp-FEM. Note, however, that standard adaptive hp-FEM approximates all three fields u1,u2 and T on the same mesh, and thus it treats T as if it were singular at re-entrant corners as well. Therefore, a large number of degrees of freedom of temperature are wasted. This motivates us to approximate the fields u1,u2 and T on individual hp-meshes equipped with mutually independent hp-adaptivity mechanisms. In this paper we describe mathematical and algorithmic aspects of the novel adaptive multimesh hp-FEM, and demonstrate numerically that it performs better than the standard adaptive h-FEM and hp-FEM.

Space and Time Adaptive Two-Mesh hp-Finite Element Method for Transient Microwave Heating Problems

Abstract A novel, highly efficient and accurate space and time adaptive higher-order finite element method (hp-FEM) is proposed for evolutionary microwave heating problems. Since the electric field E and temperature field T are very different in nature, they are approximated on individual meshes that change dynamically in time, independently of each other. Although the approximations of E and T are defined on different meshes, the coupling is treated in a monolithic fashion using a complex-valued approximate temperature. Numerical experiments are presented that show the novel method is clearly superior to its natural competitors—the space and time adaptive (single mesh) hp-finite element method and the space and time adaptive two-mesh h-finite element method. In all cases, comparisons in both the number of degrees of freedom (discrete problem size) and CPU time are presented. The methodology is freely available on-line in the form of a general public licensed C++/Python library Hermes (http://hpfem.org/).

PDE-independent adaptive hp-FEM based on hierarchic extension of finite element spaces

We present a novel approach to automatic adaptivity in higher-order finite element methods (hp-FEM) which is free of analytical error estimates. This means that a computer code based on this approach can be used to solve adaptively a wide range of PDE problems. A posteriori error estimation is done computationally via hierarchic extension of finite element spaces. This is an analogy to embedded higher-order methods for ODE. The adaptivity process yields a sequence of embedded stiffness matrices which are solved efficiently using a simple combined direct-iterative algorithm. The methodology works equally well for standard low-order FEM and for the hp-FEM. Numerical examples are presented.

Comparison of multimesh hp-FEM to interpolation and projection methods for spatial coupling of thermal and neutron diffusion calculations

Multiphysics solution challenges are legion within the ?eld of nuclear reactor design and analysis. One major issue concerns the coupling between heat and neutron ?ow (neutronics) within the reactor assembly. These phenomena are usually very tightly interdependent, as large amounts of heat are quickly produced with an increase in ?ssion events within the fuel, which raises the temperature that a?ects the neutron cross section of the fuel. Furthermore, there typically is a large diversity of time and spatial scales between mathematical models of heat and neutronics. Indeed, the di?erent spatial resolution requirements often lead to the use of very di?erent meshes for the two phenomena. As the equations are coupled, one must take care in exchanging solution data between them, or signi?cant error can be introduced into the coupled problem. We propose a novel approach to the discretization of the coupled problem on di?erent meshes based on an adaptive multimesh higher-order ?nite element method (hp-FEM), and compare it to popular interpolation and projection methods. We show that the multimesh hp-FEM method is signi?cantly more accurate than the interpolation and projection approaches considered in this study.

Hermes2D, a C++ library for rapid development of adaptive hp-FEM and hp-DG solvers

In this paper we describe Hermes2D, an open-source C++ library for the development and implementation of adaptive higher-order finite element and DG solvers for partial differential equations (PDE) and multiphysics coupled PDE problems. The library is suitable for applications ranging from simple linear PDE solvers to time-dependent solvers for nonlinear multiphysics coupled problems with dynamically changing meshes. We cover several typical application scenarios, and give a brief overview of methods and algorithms that the library provides. Numerical examples are provided.

An adaptive hp-DG method with dynamically-changing meshes for non-stationary compressible Euler equations

Compressible Euler equations describing the motion of compressible inviscid fluids are typically solved by means of low-order finite volume (FVM) or finite element (FEM) methods. A promising recent alternative to these low-order methods is the higher-order discontinuous Galerkin (

An iterative adaptive finite element method for elliptic eigenvalue problems

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