### Accurate Computations On Unstructure3d Meshes

#### Abstract

The need to perform computations on irregularly distributed nets of nodes arises in

many applications of solid and fluid computational mechanics. This is specially problematic

in three dimensions. Typically, the finite element method with tetrahedral elements is used for

such purpose. However, this poses a number of problems. On one hand some elements are

considerably distorted – with eventually some null-volume elements – leading to poor

solutions. Also, in this method only h-refinement is feasible so that solution improvement

demands to refine the mesh. In this work we describe a meshless method which we designate

as Functional Integral Method (FIM) based on the use of blurred derivatives, that allows to

overcome the above mentioned difficulties. The method only requires the connectivity of each

node given by first neighbors (Voronoi cells) for discretization yielding the same structure of

non-zeros as FEM with tetrahedral elements. The matrix is nevertheless non-symmetric so

that storage and solution of the linear system increases by a factor close to two. However,

results of several numerical simulations indicate that the error is systematically much smaller

than with FEM and it is rather insensitive to node irregularity so that relation cost-benefit is

finally enhanced substantially. Also, it allows to perform p-refinement in a trivial manner by

just adding more neighbors to the local cloud of each node thus increasing the order of

interpolation. In this way the error can be further reduced without re-meshing

many applications of solid and fluid computational mechanics. This is specially problematic

in three dimensions. Typically, the finite element method with tetrahedral elements is used for

such purpose. However, this poses a number of problems. On one hand some elements are

considerably distorted – with eventually some null-volume elements – leading to poor

solutions. Also, in this method only h-refinement is feasible so that solution improvement

demands to refine the mesh. In this work we describe a meshless method which we designate

as Functional Integral Method (FIM) based on the use of blurred derivatives, that allows to

overcome the above mentioned difficulties. The method only requires the connectivity of each

node given by first neighbors (Voronoi cells) for discretization yielding the same structure of

non-zeros as FEM with tetrahedral elements. The matrix is nevertheless non-symmetric so

that storage and solution of the linear system increases by a factor close to two. However,

results of several numerical simulations indicate that the error is systematically much smaller

than with FEM and it is rather insensitive to node irregularity so that relation cost-benefit is

finally enhanced substantially. Also, it allows to perform p-refinement in a trivial manner by

just adding more neighbors to the local cloud of each node thus increasing the order of

interpolation. In this way the error can be further reduced without re-meshing

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Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**