Hierarchic modelling in elasticity by generalized \(p\)- and \(hp\)-FEM. (English) Zbl 1298.74237
Stein, Erwin (ed.), Adaptive finite elements in linear and nonlinear solid and structural mechanics. Lecture held at CISM, Udine, Italy. Wien: Springer (ISBN 3-211-26975-4/pbk). CISM Courses and Lectures 416, 195-256 (2005).
From the introduction: We begin by introducing the 5 basic \(hp\)-elements that are used for all problem classes (solids, plates, beans, shells etc). This is already the first major difference to the usual FEM – there are no specialized and often very large element libraries for plate and shell problems required in the context of \(hp\)-FEM.
Rather, 5 \(p\)-element families (triangle-/quad-/hex-/prism-/tetrahedral families) will be sufficient to discretize all problems of linear solid mechanics successfully. Here successful means that converged FE solutions with a-posteriori quantitative error control are achievable with these elements. What does “converged” mean in this context? It refers to a negligible discretization error between the numerical solution and the exact solution of a given mathematical model. It is given in the form of a variational principle of virtual work which is the point of departure for the \(hp\)-FEM. In the first part of these notes, we present the \(hp\)-FEM for linearized elasticity. The second part presents generalizations: geometric and material nonlinearities, and their formulations.
Next, we discuss the control of the modelling error, the error inherent in the mathematical model adopted as basis for the numerical scheme. Examples constitute the choice of a two-dimensional plate – or shell theory in place of 3D elasticity, i.e. the adoption of a lower dimensional working model upon which computations are based. Converged numerical solutions can not reduce the dimension-reduction idealization error inherent in the two dimensional plate/shell theory. We present a) modeling error estimators for certain plate and shell models and b) hierarchies of plate and shell models which allow a continuous transition between dimensionally reduced plate- and shell models and three dimensional elasticity.
Differential equations modelling plate and shell problems are, due to the presence of a small parameter, the thickness of the domain, often singularly perturbed. In the zero-thickness limit, constraints are imposed on the solutions. For small, positive thickness of plates and shells, the solutions exhibit boundary layers and low order FEM are prone to locking effects. High order \(hp\)-type FEM are either free of locking or their practical performance shows very low sensitivity to it, without special “tricks”, as e.g. reduced integration etc.
Particular attention will be paid to sandwich or graded materials, where the material properties depend on the transverse coordinate of the plate/ shell. We describe the derivation and implementation of a hierarchy of shell working models which allow to resolve the interlamina stresses with a number of degrees of freedom independent of the number of layers in the sandwich material. The shell models are derived by semidiscretization in the transverse direction with non-polynomial, laminate-adapted shapefunctions and Galerkin-projection (rather than asymptotic analysis). The Galerkin-orthogonality of the discretization as well as of the modeling errors is the basis for a-posteriori error estimates, either residual or duality-based ones.
Similar concepts for homogenization problems arising from lattice and cellular materials are under development.
For the entire collection see [Zbl 1110.74010].
Rather, 5 \(p\)-element families (triangle-/quad-/hex-/prism-/tetrahedral families) will be sufficient to discretize all problems of linear solid mechanics successfully. Here successful means that converged FE solutions with a-posteriori quantitative error control are achievable with these elements. What does “converged” mean in this context? It refers to a negligible discretization error between the numerical solution and the exact solution of a given mathematical model. It is given in the form of a variational principle of virtual work which is the point of departure for the \(hp\)-FEM. In the first part of these notes, we present the \(hp\)-FEM for linearized elasticity. The second part presents generalizations: geometric and material nonlinearities, and their formulations.
Next, we discuss the control of the modelling error, the error inherent in the mathematical model adopted as basis for the numerical scheme. Examples constitute the choice of a two-dimensional plate – or shell theory in place of 3D elasticity, i.e. the adoption of a lower dimensional working model upon which computations are based. Converged numerical solutions can not reduce the dimension-reduction idealization error inherent in the two dimensional plate/shell theory. We present a) modeling error estimators for certain plate and shell models and b) hierarchies of plate and shell models which allow a continuous transition between dimensionally reduced plate- and shell models and three dimensional elasticity.
Differential equations modelling plate and shell problems are, due to the presence of a small parameter, the thickness of the domain, often singularly perturbed. In the zero-thickness limit, constraints are imposed on the solutions. For small, positive thickness of plates and shells, the solutions exhibit boundary layers and low order FEM are prone to locking effects. High order \(hp\)-type FEM are either free of locking or their practical performance shows very low sensitivity to it, without special “tricks”, as e.g. reduced integration etc.
Particular attention will be paid to sandwich or graded materials, where the material properties depend on the transverse coordinate of the plate/ shell. We describe the derivation and implementation of a hierarchy of shell working models which allow to resolve the interlamina stresses with a number of degrees of freedom independent of the number of layers in the sandwich material. The shell models are derived by semidiscretization in the transverse direction with non-polynomial, laminate-adapted shapefunctions and Galerkin-projection (rather than asymptotic analysis). The Galerkin-orthogonality of the discretization as well as of the modeling errors is the basis for a-posteriori error estimates, either residual or duality-based ones.
Similar concepts for homogenization problems arising from lattice and cellular materials are under development.
For the entire collection see [Zbl 1110.74010].
