Lifetime-Oriented Structural Design Concepts- P15 potx

378 4 Methodological Implementation (•) represents all state variables ˙ u NF , u NF and spatial gradients ∇u NF . On the basis of equation (4.14) the weak form of the coupled multiphysics system δW is generated in equation (4.15) by the weighted summation of the individual weak forms δW f ,wherebyA f is introduced to adapt physical units and dimensions of coupled fields. δW(•)= NF  f=1 A f δW f (•) = 0 (4.15) 4.2.3.2 Linearized Weak Form of Coupled Balance Equations In order to prepare the weak form for the numerical solution with the Newton-Raphson scheme, the weak form is expanded in a Taylor series about the trial solution of all state variables δW(• k+1 )=δW(• k )+ΔδW (• k ) = 0 (4.16) and spatial gradients characterized by • k . ΔδW (•)=ΔδW (∇u NF , ¨ u NF , ˙ u NF , u NF )= NF  f=1 A f ΔδW f (•)=0 (4.17) The increment of weak forms is generated by summation of individual portions in terms of the increments of field variables Δu g and gradients of field variables ∇Δu g with g ∈ [1,NF]. ΔδW f (•)= NF  g=1  ∂δW f ∂∇u g ◦ ◦ Δ∇u g + ∂δW f ∂ ˙ u g ◦Δ ˙ u g + ∂δW f ∂u g ◦ Δu g  (4.18) It is worth to mention that the derivative with respect to gradient ∇u g is performed explicitly in oder to obtain an advantageous format for 4.2 Numerical Methods 379 the finite element discretization discussed in Section 4.2.4.2. The individual terms in equation (4.18) are expressed as follows: ∂δW f ∂∇u g ◦ ◦Δ∇u g =  Ω δu f ◦ ∂ ˙ Θ f ∂∇u g ◦ ◦Δ∇u g dV +  Ω δ∇u f ◦ ◦ ∂Φ f ∂∇u g ◦ ◦Δ∇u g dV ∂δW f ∂ ˙ u g ◦ Δ ˙ u g =  Ω δu f ◦ ∂ ˙ Θ f ∂ ˙ u g ◦ Δ ˙ u g dV ∂δW f ∂u g ◦ Δu g =  Ω δu f ◦ ∂ ˙ Θ f ∂u g ◦ Δu g dV +  Ω δ∇u f ◦ ◦ ∂Φ f ∂u g ◦ Δu g dV (4.19) 4.2.4 Spatial Discretization Methods Authored by Detlef Kuhl and Christian Becker Within the framework of the semdiscretization technique applied to solve durability single- and multiphysics problems, the spatial discretization is re- alized by the finite element method (see e.g. [90, 106, 223, 224, 870, 871]). The scientific and industrial oriented literature documents the broad range of applications of this method for the spatial discretization of differential equa- tions. Highly non-linear problems, stationary and transient problems as well as single- and multifield problems can by solved adopting the finite element method. 4.2.4.1 Introduction Standard finite elements for one-, two- and three-dimensional problems are us- ing low order approximations of the primary variables. The great advantage of these finite element methods is the effective formulation and implementa- tion in finite element codes, see as a particular example the popular constant strain triangle [796]. The drawback of such kind of finite elements is the low computation accuracy because of only linear ansatz functions and locking phenomena. Basically finite element solutions can be improved by reducing the element size (h-method) and increasing the polynomial degree of ansatz functions (p-method), respectively. Furthermore, special element techniques can be used to overcome the well known locking problem. In summary, two main philosophies are used in the context of structural mechanics, to obtain high quality numerical results applying the finite element method. • Low order finite element methods combined with methods to prevent lock- ing [637, 790]: 380 4 Methodological Implementation ◦ selective reduced integration [397, 649] and hourglass control [283] ◦ assumed natural strain concept [398, 247] ◦ B-bar methods [850, 789] ◦ enhanced assumed strain concept [747, 742] • Higher order finite element methods using different kind of higher order ansatz polynomials: ◦ multidimensional Lagrange polynomials [870, 452] ◦ Legendre based hierarchical one-, two- and three-dimensional shape functions [72, 717, 246] A literature review about high quality, low order finite element methods makes clear that the related element techniques are separately developed for selected applications in structural mechanics. But computational durability mechanics is characterized by manifold various underlying differential equations. Therfore, the more general higher order finite element method is presented in the following. 4.2.4.2 Generalized Finite Element Discretization of Multifield Problems The numerical analysis of non-linear multiphysics problems can be subdivided in the spatial finite element discretization, the temporal discretization and the iterative solution of the resulting non-linear algebraic equation. In the present section a detailed description of the spatial p-finite element discretization of generalized multiphysics problems is given. 4.2.4.2.1 Approximations The finite element formulation of the original and the linearized weak forms of multiphysics problems is based on the approximation of test functions, state variables and their gradients by shape functions and nodal values of state variables. Therefore, ND -dimensional anisotropic shape functions of arbitrary polynomial degrees p d for the ND spatial directions d are designed based on one-dimensional Lagrange shape functions. Furthermore, the approximation of state variables is given and the transformation between natural and physical element coordinates is performed by the Jacobi transformation. One-dimensional Lagrange shape functions of polynomial degrees p d can be generated for every spatial direction d∈[1,ND] by the product N i (ξ d )= p d +1  k=1 k=i ξ k d − ξ d ξ k d − ξ i d ξ i d = 2[i − 1] p d − 1 (4.20) in terms of the natural coordinate ξ d and the natural nodal positions ξ i d and ξ k d with i, k ∈ [1,p d + 1]. Consequently, the derivatives of shape functions are also calculated for arbitrary polynomial degrees p d . 4.2 Numerical Methods 381 N i ;d (ξ d )= ∂N i (ξ d ) ∂ξ d = p d +1  l=1 l=i −1 ξ l d − ξ i d p d +1  k=1 k=i k=l ξ k d − ξ d ξ k d − ξ i d (4.21) ND -dimensional isotropic or anisotropic shape functions are generated by the product of one-dimensional shape functions N i (ξ)= ND  d=1 N k d (ξ d ) i = k 1 + ND  d=2 [k d −1] d−1  l=1 [p l +1] (4.22) and their derivatives with respect to natural coordinates ξ m with m∈[1,ND] analog by the product of the derivatives N k d ;m . N i ;m (ξ)=N k d ;m (ξ m ) ND  d=1 d=m N k d (ξ d ) (4.23) In equations (4.22) and (4.23) the counter k d ∈[1,p d + 1] for one-dimensional shape functions is used. Representative examples for two-dimensional isotropic and anisotropic Lagrange shape functions as well as partial derivatives of Lagrange shape functions are given in Figures 4.4 and 4.5. Furthermore, the general design of one-, two- and three-dimensional shape functions is il- lustrated in Table 4.1. Shape functions (4.22) and and spatial derivatives of shape functions (4.23) are applied for the approximation of state variables shape function N 11 derivative N 11 ;1 N 4 2 (ξ 2 ) N 2 1 (ξ 1 ) N 11 (ξ 1 ,ξ 2 ) ξ 2 ξ 1 ξ 2 ξ 1 N 4 2 (ξ 2 ) N 2 1;1 (ξ 1 ) N 11 ;1 (ξ 1 ,ξ 2 ) ξ 2 ξ 1 ξ 1 Fig. 4.4. Illustration of isotropic Lagrange shape functions and derivatives of Lagrange shape functions by means of a cubic planar finite element 382 4 Methodological Implementation shape function N 8 derivative N 8 ;2 N 2 2 (ξ 2 ) N 2 1 (ξ 1 ) N 8 (ξ 1 ,ξ 2 ) ξ 2 ξ 1 ξ 2 ξ 1 N 2 2;2 (ξ 2 ) N 2 1 (ξ 1 ) N 8 ;2 (ξ 1 ,ξ 2 ) ξ 2 ξ 1 ξ 1 Fig. 4.5. Illustration of anisotropic Lagrange shape functions and derivatives of Lagrange shape functions by means of a cubic-linear planar finite element Table 4.1. Multi-dimensional Lagrange shape functions and specialization to one-, two- and three-dimensional finite elements element type position vector X field variable u integral  Ω e • dV ND =1 truss element ND =2 plane element ND =3 volume element X =  X 1  X =  X 1 X 2  X = ⎡ ⎢ ⎣ X 1 X 2 X 3 ⎤ ⎥ ⎦ u =  u 1  u =  u 1 u 2  u = ⎡ ⎢ ⎣ u 1 u 2 u 3 ⎤ ⎥ ⎦ 1  −1 • dξ 1 A   dV ξ 1  −1 1  −1 • dξ 1 dξ 2 H    dV ξ 1  −1 1  −1 1  −1 • dξ 1 dξ 2 dξ 3    dV ξ u f ≈ NN  i=1 N i u ei f ˙ u f ≈ NN  i=1 N i ˙ u ei f (4.24) and the gradient of state variables ∇u f ≈ NN  i=1 u ei f ⊗∇N i ∇N i (ξ)= ∂N i ∂X (4.25) 4.2 Numerical Methods 383 in terms of nodal state variables u ei f and ˙ u ei f . Furthermore, equations (4.24) and (4.25) are used for the approximation of δu f , Δu f , δ∇u f and Δ∇u f δu f (ξ) ≈ NN  i=1 N i (ξ) δu ei f Δu f (ξ) ≈ NN  i=1 N i (ξ) Δu ei f δ∇u f (ξ) ≈ NN  i=1 δu ei f ⊗∇N i (ξ) Δ∇u f (ξ) ≈ NN  i=1 Δu ei f ⊗∇N i (ξ) (4.26) as well as for the implicit approximation of resulting terms ˙ Θ f , Φ f and their derivatives. Finally, the Jacobi tensor J(ξ) ≈ NN  i=1 X ei ⊗∇ ξ N i (ξ) ∇ ξ N i (ξ)= ∂N i (ξ) ∂ξ (4.27) allows for the transformation of the gradient of shape functions ∇N i (ξ)= ∂N i (ξ) ∂ξ · ∂ξ ∂X = J −T (ξ) ·∇ ξ N i (ξ) (4.28) and the differential volume element. dV = |J| dV ξ ,dV ξ = ⎧ ⎨ ⎩ dξ 1 dξ 2 dξ 3 for ND =3 dξ 1 dξ 2 H for ND =2 dξ 1 A for ND =1 (4.29) 4.2.4.2.2 Non-Linear Semidiscrete Balance Inserting the approximations discussed above into the weak form of multi- physics problems (4.14) and (4.15) for individual finite elements e ∈ [1,NE] yields the discretized weak form on the element level δW e ≈ NF  f=1 NN  i=1 δu ei f ◦  r ei if − r ei f  = 0 (4.30) in terms of the generalized internal force tensor r ei if ( ˙ u e NF , u e NF )andthe generalized external force tensor r ei f according to the nodal values of the test function δu ei f . 384 4 Methodological Implementation r ei if = A f  Ω e N i ˙ Θ f dV + A f  Ω e Φ f ·∇N i dV r ei f = A f  Ω e N i Σ dV + A f  Γ e Φ f N i φ  f dA (4.31) After assembling the element quantities over element nodes, elements and tensor fields, application of the fundamental lemma of variational cal- culus and consideration of initial values the semidiscrete initial value problem r i ( ¨ u, ˙ u, u)=ru(t 0 )=u 0 , ˙ u 0 , ¨ u 0 (4.32) is obtained. In the non-linear second order vector differential equation (4.32) the vectors r i , r, u, ˙ u and ¨ u represent the generalized vectors of internal forces, external forces, primary variables, first and second temporal rates of primary variables, respectively, whereby every structural vector contains the nodal values of all contributing fields f . As particular example the assembling of the generalized vector of internal forces r i basedonthe nodal element internal force tensors r ei i is given. r i ( ¨ u, ˙ u, u)= NE,NN  e,i r ei i ( ˙ u e NF , u e NF ) (4.33) It is worth to mention that the second time derivative only exists, if the temporal change of the balance quantity ˙ Θ f is identified with a second order time derivative as for example for the modeling of structural dynamics or wave propagation problems (see e.g [452]). The application of the present generalized multiphysics finite element con- cept for the discretization of the chemo-mechanical damage model yields the element tensors of internal and external forces (r ei i1 and r ei 1 ) and internal and external calcium ion mass fluxes (r ei i2 and r ei 2 ). r ei i1 =  Ω ∇N i ·σ dV r ei i2 =−  Ω ∇N i · qdV +  Ω N i [[φ 0 +φ 2 ]c+s]˙dV r ei 1 =  Γ σ N i t  dA r ei 2 =  Γ q N i q  dA (4.34) 4.2 Numerical Methods 385 4.2.4.2.3 Linearized Semidiscrete Balance Applying the approximation procedure to the linearized weak form (4.17-4.19) on the element level yields its discrete counterpart ΔδW e ≈ NF  f=1 NF  g=1 NN  i=1 NN  j=1 δu ei f ◦  d eij fg ◦ Δ ˙ u ej g + k eij fg ◦Δu ej g  (4.35) in terms of generalized tangent stiffness tensors k eij fg ( ˙ u e NF , u e NF )and generalized tangent damping tensors d eij fg ( ˙ u e NF , u e NF ) according to the test functions δu ei f and increments Δu ej g . k eij fg = A f  Ω N i ∂ ˙ Θ f ∂∇u g ·∇N j dV + A f  Ω ∇N i · ∂Φ f ∂∇u g ·∇N j dV + A f  Ω N i ∂ ˙ Θ f ∂u g N j dV + A f  Ω ∇N i · ∂Φ f ∂u g N j dV d eij fg = A f  Ω N i ∂ ˙ Θ f ∂ ˙ u g N j dV (4.36) On the structural level the linearized discrete balance equation is ob- tained accordingly: M( ¨ u, ˙ u, u)Δ ¨ u + D( ¨ u, ˙ u, u)Δ ˙ u + K( ¨ u, ˙ u, u)Δu = r −r i ( ¨ u, ˙ u, u) (4.37) Herein M( ¨ u, ˙ u, u), D( ¨ u, ˙ u, u)andK( ¨ u, ˙ u, u) are the generalized tan- gents which can be defined by the G ˆ ateaux derivative of the generalized internal force vector with respect to the state variables u, ˙ u and ¨ u. K( ¨ u, ˙ u, u)= ∂r i ( ¨ u, ˙ u, u) ∂u D = ∂r i ∂ ˙ u , M = ∂r i ∂ ¨ u (4.38) As particular example the generalized tangent tensors associated with the chemo-mechanical model of coupled mechanical damage and calcium leaching are given. The generalized tangent stiffness tensors are given in terms of the second order mechanical tangent stiffness tensor, 386 4 Methodological Implementation k eij 11 =  Ω ∇N i · ∂σ ∂ε ·∇N j dV (4.39) the scalar valued chemical tangent k eij 22 = −  Ω ∇N i ·  ∂q ∂c  c N j dV +  Ω ∇N i · D 0 φ ·∇N j dV +  Ω N i ∂s ∂c ∂φ 2 ∂s 2˙cN j dV +  Ω N i ∂ 2 s ∂κ 2 2 ∂κ c ∂c φ s ˙cN j dV (4.40) and the first order chemo-mechanical tangent coupling tensors. k eij 12 =  Ω ∇N i · ∂σ ∂c N j dV, k eij 21 = −  Ω ∇N i · ∂q ∂ε ·∇N j dV (4.41) Furthemore, the damping tensor of the chemo-mechanical model of calcium leaching is computed. d eij 22 =  Ω N i ∂s ∂c φ s N j dV +  Ω N i  φ 0 + φ 2  N j dV (4.42) Because of the time independent balance of momentum (4.7) the tensors d eij 11 , d eij 12 and d eij 21 vanish. In equations (4.40) and (4.42) the following abbreviations are used:  ∂q ∂c  c =[1−d] ∂φ 2 ∂c D 0 ·γ + φ ∂D 0 ∂c · γ,φ s =1+ ∂φ 2 ∂s c (4.43) 4.2.4.2.4 Generation of Element and Structural Quantities The integrals in equations (4.31) and (4.36) are computed by the Gauß- Legendre quadrature with NG =  ND d=1 NG d integration points ξ l d d , l d ∈ [1,NG d ] contained in vectors ξ l and the weights α l based on the one- dimensional integration rule, see e.g. [870].  Ω ξ f(ξ)|J(ξ)|dV ξ ≈ NG 1  l 1 =1 NG 2  l 2 =1 NG 3  l 3 =1 α l f(ξ l )|J(ξ l )|,α l = ND  l d =1 α l d (4.44) Figure 4.6 illustrates the final calculation of element quantities of generalized multiphysics p-finite elements. It is obvious that specific multiphysics problems as described in Chapter 3 can be implemented on the model or Gauß point level marked within the algorithmic set-up. Internal variables κ f are just managed on the finite element level. Manipulations of internal variables take exclusively part on the material point level. 4.2 Numerical Methods 387 ND loops over Gauss points l d ∈ [1,NG d ] next Gauß point l d loop over element nodes i ∈ [1,NN] next node i loop over element nodes i ∈ [1,NN] next node i loop over element nodes i ∈ [1,NN] next node i loop over element nodes j ∈ [1,NN] next node j select element nodal values for i ∈ [1,NN] X ei , u ei f , ˙ u ei f coordinates and weight of Gauß point ξ =[ξ l 1 1 ξ l 2 2 ξ l 3 3 ] T , α= α l 1 α l 2 α l 3 shape functions and natural derivatives N i (ξ), ∇ ξ N i (ξ) Jacobi transformation tensor J(ξ) Jacobi determinant and inverse Jacobi tensor |J(ξ)|, J −1 (ξ) state variables X(ξ), u f (ξ), ˙ u f (ξ) physical gradient of shape functions ∇N i (ξ)=J −T (ξ)·∇ ξ N i (ξ) physical gradient of tensor fields ∇u f model level ˙ Θ f ,Φ f ,∂ ˙ Θ f /∂ ∇u g ,∂ ˙ Θ f /∂ u g ,∂ ˙ Θ f /∂ ˙ u g ,∂Φ f /∂ ∇u g , generalized external load vector r ei f generalized internal force vector r ei if summation and assembly α|J|r ei f → r, α|J |r ei if → r i generalized tangent damping tensor d eij fg generalized tangent stiffness tensor k eij fg summation and assembly α|J|d eij fg → D, α|J|k eij fg → K Fig. 4.6. Computation of generalized element tensors of external and internal forces and generalized tangent damping and stiffness tensors of multiphysics p-finite ele- ments 4.2.4.3 p-Finite Element Method The p-finite element method (p-FEM)istheexactcounterparttotheh-finite element method (h-FEM). Whereas in the h-FEM a mesh of low-order elements is refined by increasing the number of elements that are naturally [...]... because they are a product of higher-order polynomials only In some formulations the latter property is used to eliminate these nodes at the structural level by a static condensation technique to achieve a better condition of the matrices and a better performance at the structural level [245] The number of nodes within each mode class depends on the order of polynomial degree in the natural coordinates and... numerical solution is not changed when using a spatially anisotropic approximation the efficiency of this method is indicated by the ratio of the total number of system nodes/dof The structural efficiency and the combined fieldwise and structural efficiency for a truss structure is Effu struc = u NNp,1,1 u NNp,p,p Effu,θ = TH u θ 3NNp,1,1 + NNp−1,1,1 , u 4NNp,p,p (4.65) whereas Taylor-Hood-like approximations were... beam or slab and 4.2 Numerical Methods 397 shell elements) Additionally, there are assumptions concerning kinetics or kinematics that aim towards a further idealization of the mechanical behaviour of the structural element Well-known representatives of these element types are Timoshenko- or Bernoulli-formulations for beams and MindlinReissner- or Kirchhoff-Love-formulations for plates On the other hand... a major drawback regarding standard higher-order elements is the considerable increase of element nodes At that point the benefit of anisotropic ansatz functions becomes clear In order to simulate the structural element in a more efficient way only the characteristic deformation behaviour is approximated by higher-order approximations Speaking in the terms of [126], discretizing only the relevant field... the in-plane approximation is higher than the approximation 398 4 Methodological Implementation moisture cracks mechanical load Θ, u, pl u temperature Θ, pl Fig 4.14 Hygro-thermo-mechanical loading of a structural segment (left), Fieldwise anisotropic discretization using the p-finite element method (right) in thickness direction Considering the environmental fields it can be vice versa Mostly shell structures... 3D-pformulation 396 4 Methodological Implementation l > (b, h) > ξ1 up,1,1 truss ξ2 ξ1 h < (l, b) < slab up,p,1 ξ2 ξ1 h < (l, b) < plate up,p,2 / up,p,3 ξ2 ξ1 shell up,p,2 / up,p,3 compact bodies up1,p2,p3 structural level classical FEM 3D-p-discretization Fig 4.13 Structure types and corresponding finite element models of the classical finite element approach and using 3D-p-elements with spatially anisotropic... full ansatz space it is: u NNp,p,p = 2n2 + 5n2 [p − 1] + 4n2 [p − 1]2 + n2 [p − 1]3 (4.70) The anisotropic approximation for the thermal field needs only Θ u NN1,1,pθ = NN1,1,pθ (4.71) nodes The pure structural efficiency can be calculated to Effu struc = u NNp,p,ps u NNp,p,p (4.72) The Taylor-Hood-discretizations are investigated for an approximation degree of ps in thickness direction and the necessary... u θ 3NNp,p,ps + NN1,1,pθ u 4NNpθ ,pθ ,pθ Effu,θ = TH u θ 3NNp,p,pθ +1 + NN1,1,pθ u 4NNpθ +1,pθ +1,pθ +1 (4.73) Figure 4.16 shows the aforementioned relations of system nodes/dof for different types of structural discretizations Regarding aspects of efficiency, it is stressed here, that the number of system nodes or dof is just an indicator for the real reduction of computing time In fact, the reduction... powerful tool is developed in conjunction with the p-finite element method [246] Despite all the efficiency there is a lot of extra work to be done concerning the generation of the connectivities between structural and geometrical entities Therefore in the presented formulation a subparametric concept is used The geometry is described by a twenty-noded continuum element which allows for a quadratic Serendipity . eliminate these nodes at the structural level by a static condensation technique to achieve a better condition of the matrices and a better perfor- mance at the structural level [245]. The number. makes clear that the related element techniques are separately developed for selected applications in structural mechanics. But computational durability mechanics is characterized by manifold various. anisotropic shape functions of arbitrary polynomial degrees p d for the ND spatial directions d are designed based on one-dimensional Lagrange shape functions. Furthermore, the approximation of state