Stabilized plane and axisymmetric Lobatto finite element models

Abstract

High order elements are renowned for their high accuracy and convergence. Among them, Lobatto spectral finite elements are commonly used in explicit dynamic analyses as their mass matrices when evaluated by the Lobatto integration rule are diagonal. While there are numerous advanced first and second order elements, advanced high order elements are rarely seen. In this paper, generic stabilization schemes are devised for the reduced integrated plane and axisymmetric elements. Static and explicit dynamic tests are considered for evaluating the relatively merits of the stabilized and conventional elements. The displacement errors of the stabilized elements are less than those of the conventional Lobatto elements. When the material is nearly incompressible, the stabilized elements are also more accurate in terms of the energy error norm. This advantage is of practical importance for bio-tissue and hydrated soil analyses.

Appendix: Assembling the element stiffness matrices

In this appendix, the LB and URI element stiffness matrices formed by the assembling procedure is presented. By recalling the property $L_{i}(l_{j}) = \delta _{ij}$, the derivatives for any interpolated variable $\phi$ at $(l_{k},l_{l})$ with respect to the parametric coordinates are

$$\begin{aligned}&\left. {\phi ,_\xi }\right| _{(\xi =l_k ,\eta =l_l )} \\&\qquad =\sum _{i,j=1}^{n+1} {{L}'_i (l_k )L_j (l_l )\phi _{ij} } =\sum _{i=1}^{n+1} {{L}'_i (l_k )\phi _{il} } \\&\qquad =\left[ {L}'_1 (l_k ),\ldots ,{L}'_n (l_k )\right] \left\{ {{\begin{array}{l} {\phi _{1l} } \\ \vdots \\ {\phi _{nl} } \\ \end{array} }} \right\} \\&\qquad =\left[ {L}'_1 (l_k ),\ldots ,{L}'_n (l_k )\right] \cdot \underline{\phi _{Ql} }, \quad \left. {\phi ,_\eta } \right| _{(\xi =l_k ,\eta =l_l )}\\&\qquad =\sum _{i,j=1}^{n+1} {L_i (l_k ){L}'_j (l_l )\phi _{ij} }\\&\qquad =\sum _{j=1}^{n+1} {{L}'_j (l_l )\phi _{kj} } =\left[ {L}'_1 (l_l ),\ldots ,{L}'_n (l_l )\right] \left\{ {{\begin{array}{l} {\phi _{k1} } \\ \vdots \\ {\phi _{kn} } \\ \end{array} }} \right\} \\&\qquad =\left[ {L}'_1 (l_l ),\ldots ,{L}'_n (l_l )\right] \cdot \underline{\phi _{Pk}} \end{aligned}$$

in which $\underline{{\phi }_{Pi}}$ and $\underline{{\phi }_{Qi}}$ are respectively the vectors containing the nodal values of $\phi$ in the following Lobatto node sets:

$$\begin{aligned} Pi= & {} \{(l_i ,l_1 ),(l_i ,l_2 ),\ldots ,(l_i ,l_n ),(l_i ,l_{n+1} )\}\\&\quad \hbox { for}\,i = 1,\ldots ,n+1;\\ Qi= & {} \{(l_1 ,l_i ),(l_2 ,l_i ),\ldots ,(l_n ,l_i ),(l_{n+1} ,l_i )\}\\&\quad \hbox { for}\,i = 1,\ldots ,n+1, \end{aligned}$$

see Fig. 2b for illustration. Moreover, ${L}'_i (l_j )$ can be pre-computed as in the spectral and differential quadrature methods, see [35] among others. Through the chain rule of differentiation

$$\begin{aligned}&\frac{\partial }{\partial x}=\frac{1}{J}\left( \frac{\partial y}{\partial \eta }\frac{\partial }{\partial \xi }-\frac{\partial y}{\partial \xi }\frac{\partial }{\partial \eta }\right) ,\\&\quad \frac{\partial }{\partial y}=\frac{1}{J}\left( -\frac{\partial x}{\partial \eta }\frac{\partial }{\partial \xi }+\frac{\partial x}{\partial \xi }\frac{\partial }{\partial \eta }\right) \end{aligned}$$

where $J = (\partial x/\partial \xi )(\partial y/\partial \eta )-(\partial y/\partial \xi )(\partial x/\partial \eta )$ is the Jacobian determinant, the strain component at the Lobatto node (i, j) can be expressed as:

$$\begin{aligned} \underline{\varepsilon _{ij} }= & {} \left. {\left\{ {{\begin{array}{l} {\varepsilon _x } \\ {\varepsilon _y } \\ {\gamma _{xy} } \\ \end{array} }} \right\} } \right| _{(\xi =l_i ,\eta =l_j )} =\left. {\left\{ {{\begin{array}{l} {u,_x } \\ {v,_y } \\ {u,_y +v,_x } \\ \end{array} }} \right\} } \right| _{(\xi =l_i ,\eta =l_j )}\\= & {} \underline{\underline{B_{PiQj}}} \underline{d_{PiQj} } \end{aligned}$$

where $\underline{\underline{B_{PiQj}}}$ is the strain-displacement matrix for node (i, j) with respect to ${\underline{d_{PiQj}}}$ which is vector containing the $2(2n+1)$ nodal displacement components in the node sets Pi and Qj. Using the LBQ rule in which the integration points are the Lobatto nodes, the strain energy in the element can be expressed as:

$$\begin{aligned}&\frac{1}{2}\left\langle \underline{\varepsilon }^{T}\underline{\underline{C}} \underline{\varepsilon }\right\rangle _{{ LBQ}}^e =\frac{1}{2}\sum _{i,j=1}^{n+1} {\left( w_i w_j J_{ij} \underline{\varepsilon _{ij}^T }{\underline{\underline{C}}} \underline{\varepsilon _{ij}}\right) } \\&\quad =\frac{1}{2}\sum _{i,j=1}^{n+1} {\left( w_i w_j J_{ij} \underline{d_{PiQj}^T }\underline{\underline{B_{PiQj}^T }} {\underline{\underline{C}}} \underline{\underline{B_{PiQj}}} \underline{d_{PiQj}}\right) } \\&\quad =\frac{1}{2}\underline{d}^T \underline{\underline{k_{{ LB}}}} \underline{d} \end{aligned}$$

where $w_{i}$ is the weight factor for the Lobatto node $l_{i},J_{ij} =\left. J \right| _{(\xi =l_i ,\eta =l_j )}$. As dim.($\underline{\underline{B_{PiQj}^T }} {\underline{\underline{C}}} \underline{\underline{B_{PiQj}}}) = 2(2n+1)\times 2(2n+1)$ is smaller or much smaller than dim. $(\underline{\underline{k_{{ LB}}}}) = 2(n+1)^{2}\times 2(n+1)^{2}$, the process of forming $\underline{\underline{k_{{ LB}}}}$ from $\underline{\underline{B_{PiQj}^T }} \underline{\underline{C}} \underline{\underline{B_{PiQj}}}$ can be speeded up by using matrix assembling which is similar to that used in forming the system matrix from the element matrices.

For the URI element, the derivatives for any variable $\phi$ at RGQ point $(g_{k},g_{l})$ with respect to the parametric coordinates are

$$\begin{aligned}&\left. {\phi ,_\xi } \right| _{\xi =g_k ,\eta =g_l } =\sum _{i,j=1}^{n+1} {{L}'_i (g_k )L_j (g_l )\phi _{ij} }, \nonumber \\&\quad \left. {\phi ,_\eta } \right| _{\xi =g_k ,\eta =g_l } =\sum _{i,j=1}^{n+1} {L_i (g_k ){L}'_j (g_l )\phi _{ij} } \end{aligned}$$

(50)

As

$$\begin{aligned}&\left. \phi \right| _{\xi =l_i ,\eta =g_l } =\sum _{m,j=1}^{n+1} {L_m (l_i )L_j (g_l )\phi _{ij} } =\sum _{j=1}^{n+1} {L_j (g_l )\phi _{ij} } \quad \hbox { and}\nonumber \\&\quad \left. \phi \right| _{\xi =g_k ,\eta =l_j } =\sum _{i,m=1}^{n+1} {L_i (g_k )L_m (l_j )\phi _{im} } =\sum _{i=1}^{n+1} {L_i (g_k)\phi _{ij}},\nonumber \\ \end{aligned}$$

(51)

Eq. (50) can be expressed as

$$\begin{aligned} \left. {\phi ,_\xi } \right| _{\xi =g_k ,\eta =g_l }= & {} \sum _{i=1}^{n+1} {{L}'_i (g_k )\left. \phi \right| _{\xi =l_i ,\eta =g_l } } \\= & {} \left[ {L}'_1 (g_k ),\ldots , {L}'_n (g_k )\right] \left\{ {{\begin{array}{l} {\left. \phi \right| _{\xi =l_1 ,\eta =g_l } } \\ \vdots \\ {\left. \phi \right| _{\xi =l_n ,\eta =g_l } } \\ \end{array} }} \right\} \\= & {} \left[ {L}'_1 (g_k ),\ldots ,{L}'_n (g_k )\right] \cdot \underline{\phi _{{\mathop {Q}\limits ^{\frown }} l} },\\ \left. {\phi ,_\eta } \right| _{\xi =g_k ,\eta =g_l }= & {} \sum _{j=1}^{n+1} {{L}'_j (g_l )\left. \phi \right| _{\xi =g_k ,\eta =l_j } }\\= & {} \left[ {L}'_1 (g_l ),\ldots ,{L}'_n (g_l )\right] \left\{ {{\begin{array}{l} {\left. \phi \right| _{\xi =g_k ,\eta =l_1 } } \\ \vdots \\ {\left. \phi \right| _{\xi =g_k ,\eta =l_n } } \\ \end{array} }} \right\} \\= & {} \left[ {L}'_1 (g_l ),\ldots ,{L}'_n (g_l )\right] \cdot \underline{\phi _{{\mathop {P}\limits ^{\frown }} k} } \end{aligned}$$

It can be seen that ${\mathop {P}\limits ^{\frown }} i$ and ${\mathop {Q}\limits ^{\frown }} i$ are respectively the vectors containing the values of the following auxiliary node sets:

$$\begin{aligned} {\mathop {P}\limits ^{\frown }} i= & {} \{(g_i ,l_1 ),(g_i ,l_2 ),\ldots ,(g_i ,l_n ),(g_i ,l_{n+1} )\}\nonumber \\&\mathrm{for}\, i = 1,\ldots ,n,\\ {\mathop {Q}\limits ^{\frown }} i= & {} \{(l_1 ,g_i ),(l_2 ,g_i ),\ldots ,(l_n ,g_i ),(l_{n+1} ,g_i )\}\nonumber \\&\mathrm{for}\, i = 1,\ldots ,n. \end{aligned}$$

see Fig. 2c for illustration. Again, ${L}'_i (g_j )$ can be pre-computed. Similar to the standard Lobatto element, the strain components at each RGQ point and the strain energy of the element can be expressed as:

$$\begin{aligned} \underline{{\mathop {\varepsilon }\limits ^{\frown }}_{ij} }= & {} \left. {\left\{ {{\begin{array}{l} {\varepsilon _x} \\ {\varepsilon _y} \\ {\gamma _{xy} } \\ \end{array} }} \right\} } \right| _{(\xi =g_i ,\eta =g_j )} =\left. {\left\{ {{\begin{array}{l} {u,_x } \\ {v,_y } \\ {u,_y +v,_x } \\ \end{array} }} \right\} } \right| _{(\xi =g_i ,\eta =g_j )} \nonumber \\= & {} \underline{\underline{\mathop {\mathop {B}\limits ^{\frown }}\nolimits _{{\mathop {P}\limits ^{\frown }} i{\mathop {Q}\limits ^{\frown }} j} }} \underline{d_{{\mathop {P}\limits ^{\frown }} i{\mathop {Q}\limits ^{\frown }} j} }, \end{aligned}$$

$$\begin{aligned} \frac{1}{2}\left\langle \underline{\varepsilon }^{T}\underline{\underline{C}} \underline{\varepsilon }\right\rangle _{{ RGQ}}^e= & {} \frac{1}{2}\sum _{i,j=1}^n {\left( \mathop {\mathop {w}\limits ^{\frown }}\nolimits _i \mathop {\mathop {w}\limits ^{\frown }}\nolimits _j \mathop {\mathop {J}\limits ^{\frown }}\nolimits _{ij} \underline{{\mathop {\varepsilon }\limits ^{\frown }}_{ij}^T }{\underline{\underline{C}}} \underline{{\mathop {\varepsilon }\limits ^{\frown }}_{ij}}\right) } \nonumber \\= & {} \frac{1}{2}\sum _{i=1}^n {\left( \mathop {\mathop {w}\limits ^{\frown }}\nolimits _i \mathop {\mathop {w}\limits ^{\frown }}\nolimits _j \mathop {\mathop {J}\limits ^{\frown }}\nolimits _{ij} \underline{d_{{\mathop {P}\limits ^{\frown }} i{\mathop {Q}\limits ^{\frown }}j}^T }\underline{\underline{{\mathop {B}\limits ^{\frown }} _{{\mathop {P}\limits ^{\frown }} i{\mathop {Q}\limits ^{\frown }}j}^T }} {\underline{\underline{C}}} \underline{\underline{\mathop {\mathop {B}\limits ^{\frown }}\nolimits _{{\mathop {P}\limits ^{\frown }} i{\mathop {Q}\limits ^{\frown }} j}}} \underline{d_{{\mathop {P}\limits ^{\frown }} i{\mathop {Q}\limits ^{\frown }}j}}\right) } \nonumber \\= & {} \frac{1}{2}\underline{{\mathop {d}\limits ^{\frown }}}^T \underline{\underline{{\mathop {k}\limits ^{\frown }}}} \underline{{\mathop {d}\limits ^{\frown }}} \end{aligned}$$

(52)

where $\underline{\underline{{\mathop {B}\limits ^{\frown }} _{{\mathop {P}\limits ^{\frown }} i{\mathop {Q}\limits ^{\frown }} j}}}$ is the strain-displacement matrix for the RGQ point $(g_{i},g_{j})$ with respect to $\underline{d_{{\mathop {P}\limits ^{\frown }} i{\mathop {Q}\limits ^{\frown }}j}}$ which is vector containing the $4(n+1)$ displacement components in the auxiliary node sets ${\mathop {P}\limits ^{\frown }} i$ and ${\mathop {Q}\limits ^{\frown }} j,\mathop {\mathop {J}\limits ^{\frown }}\nolimits _{ij} =\left. J \right| _{(\xi =g_i ,\eta =g_j )}$ and $\underline{{\mathop {d}\limits ^{\frown }} }$ is vector containing the $4n(n+1)$ displacement components in the auxiliary node sets ${\mathop {P}\limits ^{\frown }} i$ and ${\mathop {Q}\limits ^{\frown }} i$ for $i = 1,..,n$. Again, $\underline{\underline{{\mathop {k}\limits ^{\frown }}}}$ can be assembled from $\underline{\underline{{\mathop {B}\limits ^{\frown }} _{{\mathop {P}\limits ^{\frown }}i{\mathop {Q}\limits ^{\frown }}j}^T}} {\underline{\underline{C}}} \underline{\underline{\mathop {\mathop {B}\limits ^{\frown }}\nolimits _{{\mathop {P}\limits ^{\frown }} i{\mathop {Q}\limits ^{\frown }} j}}}$. To derive the element stiffness matrix, the relation between the nodes and auxiliary nodes can be expressed as:

$$\begin{aligned} \underline{{\mathop {d}\limits ^{\frown }} }=\underline{\underline{T_a}} \underline{d} \end{aligned}$$

(53)

in which $\underline{\underline{T_{a}}}$ is a sparse transformation matrix with $n+1$ non-zero entries per row. From (52) and (53), the stiffness matrix of the URI element is

$$\begin{aligned} \underline{\underline{k_{{ URI}}}} =\underline{\underline{T_a}}^T \underline{\underline{{\mathop {k}\limits ^{\frown }}}} \underline{\underline{T_a}}. \end{aligned}$$

(54)