PSI - Issue 52

Tong-Rui Liu et al. / Procedia Structural Integrity 52 (2024) 740–751 T.-R. Liu, F. Aldakheel, M. H. Aliabadi / Structural Integrity Procedia 00 (2023) 000–000

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Figure 2: Schematic description of a 2D polygonal element E , x c is the centroid of E

  displacement: V h : =  v h ∈V : v h | E ∈V h | E , ∀ E ∈ Ω h  phase field: Q h : =  q h ∈Q : q h | E ∈Q h | E , ∀ E ∈ Ω h 

(8)

where V h | E and Q h | E represent the local virtual space for displacement and phase field, respectively. Following the work of Beira˜o da Veiga (2013); Gain (2014), a lower order VEM formulation (with linear polynomial) is taken into consideration. For E ∈ Ω h , the discrete virtual space Q h | E at element level is defined as: Q h | E : =  q ∈  H 1 ( E ) ∩ C 0 ( E )  : ∆ q = 0 in E , q | e ∈P 1 ( e ) , ∀ e ∈ ∂ E  (9) Q h | E is scalar value-based functional space for the phase field variable d , while the extension towards vectorial space for displacement field u is straightforward, such as: V h | E : =  v ∈  H 1 ( E ) ∩ C 0 ( E )  2 : ∆ v = 0 in E , v | e ∈ [ P 1 ( e )] 2 , ∀ e ∈ ∂ E  (10) ∆ represents the component-wise Laplacian operator. P 1 ( e ) represents the polynomial space of degree ≤ 1 defined on e . Q h | E and V h | E contain the continuous harmonic functions of which piece-wise linear on e and vanishing Laplacian interior E . The ‘virtual’ means that the space is well defined on the boundary of the element but not explicitly known inside the element. The spaces Q h | E and V h | E are linearly complete, suggesting that [ P 1 ( E )] 2 ⊆V h | E and [ P 1 ( E )] ⊆ Q h | E . Thus, only the degree of freedom (DOF) for V h | E and Q h | E are taken at the vertices of E . Providing that, values at DOFs are given and v ∈ V h | E and q ∈ Q h | E are linear on e , the value of v and q are completely known on the boundary of E . By virtue of the Gauss divergence theorem and integrating by part, one can