[工学]Slide_Chpt09-FEM for 3D Solids.ppt

Finite Element Method FEM FOR 3D SOLIDS CONTENTS INTRODUCTION TETRAHEDRON ELEMENT Shape functions Strain matrix Element matrices HEXAHEDRON ELEMENT Shape functions Strain matrix Element matrices Using tetrahedrons to form hexahedrons HIGHER ORDER ELEMENTS ELEMENTS WITH CURVED SURFACES CASE STUDY INTRODUCTION For 3D solids, all the field variables are dependent of x, y and z coordinates – most general element. The element is often known as a 3D solid element or simply a solid element. A 3-D solid element can have a tetrahedron and hexahedron shape with flat or curved surfaces. At any node there are three components in x, y and z directions for the displacement as well as forces. TETRAHEDRON ELEMENT 3D solid meshed with tetrahedron elements TETRAHEDRON ELEMENT Shape functions Shape functions Shape functions Shape functions Shape functions Strain matrix Element matrices Element matrices Element matrices Element matrices Element matrices Element matrices Element matrices Element matrices HEXAHEDRON ELEMENT 3D solid meshed with hexahedron elements Shape functions Shape functions Strain matrix Strain matrix Strain matrix Strain matrix Element matrices Element matrices Element matrices Element matrices Element matrices Element matrices Element matrices Using tetrahedrons to form hexahedrons Hexahedrons can be made up of several tetrahedrons Using tetrahedrons to form hexahedrons Element matrices can be obtained by assembly of tetrahedron elements HIGHER ORDER ELEMENTS Tetrahedron elements HIGHER ORDER ELEMENTS Tetrahedron elements (Cont’d) HIGHER ORDER ELEMENTS Brick elements HIGHER ORDER ELEMENTS Brick elements (Cont’d) HIGHER ORDER ELEMENTS Brick elements (Cont’d) ELEMENTS WITH CURVED SURFACES CASE STUDY Stress and strain analysis of a quantum dot heterostructure CASE STUDY CASE STUDY CASE STUDY CASE STUDY (Cont’d) E.g. (Cont’d) Note: For x direction only (Rectangular hexahedron) 1 7 5 8 6 4 2 0 z y x 3 0 fsz fsy fsx For uniformly distributed load: Hexahedron made up