The Long Range Elastic Energy � � E elastic ( u ) = ( u ( x ) − u ( y )) T J ( x − y )( u ( x ) − u ( y )) dx dy Q Q The kernel J depends on the crystal lattice and satisfies the following properties: Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
The Long Range Elastic Energy � � E elastic ( u ) = ( u ( x ) − u ( y )) T J ( x − y )( u ( x ) − u ( y )) dx dy Q Q The kernel J depends on the crystal lattice and satisfies the following properties: J ( t ) ∈ M N × N Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
The Long Range Elastic Energy � � E elastic ( u ) = ( u ( x ) − u ( y )) T J ( x − y )( u ( x ) − u ( y )) dx dy Q Q The kernel J depends on the crystal lattice and satisfies the following properties: J ( t ) ∈ M N × N J defines a positive quadratic form which is equivalent to the square 1 2 seminorm: of the H | t | 3 | ξ | 2 ≤ ξ T J ( t ) ξ ≤ c 2 c 1 | t | 3 | ξ | 2 Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
The Core Energy of Dislocations It is described by a piecewise quadratic potential which penalizes distorsions of the crystal lattice induced by slip fields u �∈ Z N . Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
The Core Energy of Dislocations It is described by a piecewise quadratic potential which penalizes distorsions of the crystal lattice induced by slip fields u �∈ Z N . E core ( u ) = 1 � dist 2 ( u ( x ) , Z N ) dx ε Q Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
The Core Energy of Dislocations It is described by a piecewise quadratic potential which penalizes distorsions of the crystal lattice induced by slip fields u �∈ Z N . E core ( u ) = 1 � dist 2 ( u ( x ) , Z N ) dx ε Q ε ∼ distance between the atoms of the crystal Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
The Core Energy of Dislocations It is described by a piecewise quadratic potential which penalizes distorsions of the crystal lattice induced by slip fields u �∈ Z N . E core ( u ) = 1 � dist 2 ( u ( x ) , Z N ) dx ε Q ε ∼ distance between the atoms of the crystal Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
The Energy Functional � � ( u ( x ) − u ( y )) T J ( x − y )( u ( x ) − u ( y )) dx dy +1 � dist 2 ( u , Z N ) dx E ε ( u ) = ε Q Q Q Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
The Energy Functional � � ( u ( x ) − u ( y )) T J ( x − y )( u ( x ) − u ( y )) dx dy +1 � dist 2 ( u , Z N ) dx E ε ( u ) = ε Q Q Q A multi-well potential functional with a non local, singular and anisotropic perturbation Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
The Energy Functional � � ( u ( x ) − u ( y )) T J ( x − y )( u ( x ) − u ( y )) dx dy +1 � dist 2 ( u , Z N ) dx E ε ( u ) = ε Q Q Q A multi-well potential functional with a non local, singular and anisotropic perturbation Crystallographic Slip ⇐ ⇒ Phase Transition Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Phase Fields dist 2 ( u ( x ) , Z N ) = ξ ( x ) ∈ Z N | u ( x ) − ξ ( x ) | 2 min Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Phase Fields dist 2 ( u ( x ) , Z N ) = ξ ( x ) ∈ Z N | u ( x ) − ξ ( x ) | 2 min ξ = ( ξ 1 , ..., ξ N ) : S → Z N Phase Field Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Phase Fields dist 2 ( u ( x ) , Z N ) = ξ ( x ) ∈ Z N | u ( x ) − ξ ( x ) | 2 min ξ = ( ξ 1 , ..., ξ N ) : S → Z N Phase Field Dislocations are identified with the jump set of ξ Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
The Energy Functional � � ( u ( x ) − u ( y )) T J ( x − y )( u ( x ) − u ( y )) dx dy +1 � dist 2 ( u , Z N ) dx E ε ( u ) = ε Q Q Q Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Γ-convergence Let ( X , d ) be a metric space and F ε , F : X → [0 , + ∞ ]. Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Γ-convergence Let ( X , d ) be a metric space and F ε , F : X → [0 , + ∞ ]. F ε Γ( d )-converges to F if Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Γ-convergence Let ( X , d ) be a metric space and F ε , F : X → [0 , + ∞ ]. F ε Γ( d )-converges to F if d For every u ∈ X and every sequence { u ε } ⊆ X such that u ε → u it follows that lim inf ε → 0 F ε ( u ε ) ≥ F ( u ) . Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Γ-convergence Let ( X , d ) be a metric space and F ε , F : X → [0 , + ∞ ]. F ε Γ( d )-converges to F if d For every u ∈ X and every sequence { u ε } ⊆ X such that u ε → u it follows that lim inf ε → 0 F ε ( u ε ) ≥ F ( u ) . d For every u ∈ X there exists { u ε } ⊆ X such that u ε → u and ε → 0 F ε ( u ε ) = F ( u ) . lim u ε is called an optimal sequence or a recovery sequence Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Rescaling of the Energy How much a phase transition ”costs”? Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Rescaling of the Energy How much a phase transition ”costs”? u = u = ε 0 s 0 s A B ε 0 , s ∈ Z N Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Rescaling of the Energy How much a phase transition ”costs”? u = u = ε 0 s 0 s A B ε 0 , s ∈ Z N | u ε ( x ) − u ε ( y ) | 2 � � E ε ( u ε ) ∼ C dx dy + l.o.t. | x − y | 3 Q Q | u ε ( x ) − u ε ( y ) | 2 � � = C dx dy + l.o.t. = C | log ε | + l.o.t. | x − y | 3 A B Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Γ-convergence of the Energy Functional 1 � � 1 � dist 2 ( u , Z N ) dx ( u ( x ) − u ( y )) T J ( x − y )( u ( x ) − u ( y )) dx dy + F ε ( u ) = | log ε | ε | log ε | Q Q Q Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Γ-convergence of the Energy Functional 1 � � 1 � dist 2 ( u , Z N ) dx ( u ( x ) − u ( y )) T J ( x − y )( u ( x ) − u ( y )) dx dy + F ε ( u ) = | log ε | ε | log ε | Q Q Q Theorem (C.-Garroni) Compactness If F ε ( u ε ) ≤ M, then ∃ a ε ∈ Z N and u ∈ BV ( Q , Z N ) such that (up to sub-sequences) in L 1 ( Q ) u ε − a ε → u Γ -convergence ∃ a sub-sequence ε k → 0 and a function ϕ : Z N × S 1 → R such that � Γ( L 1 ) -converges to ϕ ([ u ] , n u ) d H 1 F ε k ( u ) F ( u ) = Su Su = jump set of u [ u ] = jump of u n u = unit normal to Su Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Problems of the Abstract Approach Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Problems of the Abstract Approach The functional F depends on the sub-sequence ε k Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Problems of the Abstract Approach The functional F depends on the sub-sequence ε k ϕ ( s , n ) =? Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Problems of the Abstract Approach The functional F depends on the sub-sequence ε k ϕ ( s , n ) =? ∀ ( s , n ) ∈ Z N × S 1 ϕ ( s , n ) = F ( u n s , Q n ) u n s ( x ) = s χ { x · n > 0 } 0 n n u = s s Q n Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Problems of the Abstract Approach The functional F depends on the sub-sequence ε k ϕ ( s , n ) =? ∀ ( s , n ) ∈ Z N × S 1 ϕ ( s , n ) = F ( u n s , Q n ) u n s ( x ) = s χ { x · n > 0 } 0 n n u = s s Q n We do not know optimal sequences: u ε → u n ε → 0 F ε ( u ε , Q n ) = F ( u n s , Q n ) lim s Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Problems of the Abstract Approach The functional F depends on the sub-sequence ε k ϕ ( s , n ) =? ∀ ( s , n ) ∈ Z N × S 1 ϕ ( s , n ) = F ( u n s , Q n ) u n s ( x ) = s χ { x · n > 0 } 0 n n u = s s Q n We do not know optimal sequences: u ε → u n ε → 0 F ε ( u ε , Q n ) = F ( u n s , Q n ) lim s Completely solved in a scalar case (Garroni-M¨ uller) Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Scalar Case Only one slip system active Slip Field = u b where u : Q → R is a scalar function and b is a given Burgers vector. Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Scalar Case Only one slip system active Slip Field = u b where u : Q → R is a scalar function and b is a given Burgers vector. The functional reduces to 1 � � 1 � dist 2 ( u , Z ) dx J( x − y ) | u ( x ) − u ( y ) | 2 dx dy + F ε ( u , Q ) = | log ε | ε | log ε | Q Q Q Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Scalar Case Only one slip system active Slip Field = u b where u : Q → R is a scalar function and b is a given Burgers vector. The functional reduces to 1 � � 1 � dist 2 ( u , Z ) dx J( x − y ) | u ( x ) − u ( y ) | 2 dx dy + F ε ( u , Q ) = | log ε | ε | log ε | Q Q Q Theorem (Garroni-M¨ uller) � Γ( L 1 ) -converges to F ( u , Q ) = γ ( n u ) | [ u ] | d H 1 F ε ( u , Q ) Su ∩ Q where � γ ( n ) = 2 J ( x ) dx x · n =1 Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Scalar Case Optimal sequence: it sufficies to consider any mollification of u n = χ { x · n > 0 } , i.e. u ε = u n ∗ φ ε 0 n n 0 n u ε u = = 1 1 Q n Q n ε Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Scalar Case Optimal sequence: it sufficies to consider any mollification of u n = χ { x · n > 0 } , i.e. u ε = u n ∗ φ ε 0 n n 0 n u ε u = = 1 1 Q n Q n ε 1 � � Q n J( x − y ) | u ( x ) − u ( y ) | 2 dx dy = γ ( n ) = F ( u n , Q n ) ε → 0 F ε ( u ε , Q n ) = lim lim | log ε | ε → 0 Q n Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Scalar Case Optimal sequence: it sufficies to consider any mollification of u n = χ { x · n > 0 } , i.e. u ε = u n ∗ φ ε 0 n n 0 n u ε u = = 1 1 Q n Q n ε 1 � � Q n J( x − y ) | u ( x ) − u ( y ) | 2 dx dy = γ ( n ) = F ( u n , Q n ) ε → 0 F ε ( u ε , Q n ) = lim lim | log ε | ε → 0 Q n ”One-Dimensional Profile” Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Vector Case: the Cubic Lattice Two slip systems active Slip Field = u 1 b 1 + u 2 b 2 where u = ( u 1 , u 2 ) : Q → R 2 and b 1 , b 2 are two Burgers vectors parallel to the versors of the canonical basis of R 2 . Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Vector Case: the Cubic Lattice Two slip systems active Slip Field = u 1 b 1 + u 2 b 2 where u = ( u 1 , u 2 ) : Q → R 2 and b 1 , b 2 are two Burgers vectors parallel to the versors of the canonical basis of R 2 . The functional reduces to 1 � � 1 � dist 2 ( u , Z 2 ) dx ( u ( x ) − u ( y )) T J ( x − y )( u ( x ) − u ( y )) dx dy + F ε ( u , Q ) = | log ε | ε | log ε | Q Q Q Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Vector Case: the Cubic Lattice Two slip systems active Slip Field = u 1 b 1 + u 2 b 2 where u = ( u 1 , u 2 ) : Q → R 2 and b 1 , b 2 are two Burgers vectors parallel to the versors of the canonical basis of R 2 . The functional reduces to 1 � � 1 � dist 2 ( u , Z 2 ) dx ( u ( x ) − u ( y )) T J ( x − y )( u ( x ) − u ( y )) dx dy + F ε ( u , Q ) = | log ε | ε | log ε | Q Q Q s = s χ { x · n > 0 } with s ∈ Z 2 , n ∈ S 1 and Strategy of the scalar case: u n u ε = u n s ∗ φ ε 0 n 0 n u ε n u = = s s s Q n Q n ε Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Vector Case: the Cubic Lattice ε → 0 F ε ( u ε , Q n ) = s T γ ( n ) s = γ 11 ( n ) s 2 1 + γ 22 ( n ) s 2 2 +2 γ 12 ( n ) s 1 s 2 =: F flat ( u n s , Q n ) lim Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Vector Case: the Cubic Lattice ε → 0 F ε ( u ε , Q n ) = s T γ ( n ) s = γ 11 ( n ) s 2 1 + γ 22 ( n ) s 2 2 +2 γ 12 ( n ) s 1 s 2 =: F flat ( u n s , Q n ) lim n θ 2 − 2 ν sin 2 θ ν sin 2 θ � γ ( n ) = 2 J ( x ) dx = γ ( θ ) = C ν 2 − 2 ν cos 2 θ x · n =1 ν sin 2 θ Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Vector Case: the Cubic Lattice ε → 0 F ε ( u ε , Q n ) = s T γ ( n ) s = γ 11 ( n ) s 2 1 + γ 22 ( n ) s 2 2 +2 γ 12 ( n ) s 1 s 2 =: F flat ( u n s , Q n ) lim n θ 2 − 2 ν sin 2 θ ν sin 2 θ � γ ( n ) = 2 J ( x ) dx = γ ( θ ) = C ν 2 − 2 ν cos 2 θ x · n =1 ν sin 2 θ γ ( n ) is positive defined, but the second diagonal may change sign Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Vector Case: the Cubic Lattice In some directions it is better to split the jumps: Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Vector Case: the Cubic Lattice In some directions it is better to split the jumps: θ = π 4 , s = (1 , 1) (1,1) (1,1) v ε (1,0) u = = (0,0) (0,0) δε >> ε Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Vector Case: the Cubic Lattice In some directions it is better to split the jumps: θ = π 4 , s = (1 , 1) (1,1) (1,1) v ε (1,0) u = = (0,0) (0,0) δε >> ε ε → 0 F ε ( u ∗ φ ε , Q n ) = γ 11 ( θ )+ γ 22 ( θ )+2 γ 12 ( θ ) > γ 11 ( θ )+ γ 22 ( θ ) = lim ε → 0 F ε ( v ε ∗ φ ε , Q n ) lim Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Vector Case: the Cubic Lattice In some directions it is better to pile-up the jumps: Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Vector Case: the Cubic Lattice In some directions it is better to pile-up the jumps: θ = − π 4 , s = (1 , 1) δ ε ε >> (0,0) (0,0) v ε (1,0) u = = (1,1) (1,1) Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Vector Case: the Cubic Lattice In some directions it is better to pile-up the jumps: θ = − π 4 , s = (1 , 1) δ ε ε >> (0,0) (0,0) v ε (1,0) u = = (1,1) (1,1) ε → 0 F ε ( u ∗ φ ε , Q n ) = γ 11 ( θ )+ γ 22 ( θ )+2 γ 12 ( θ ) < γ 11 ( θ )+ γ 22 ( θ ) = lim ε → 0 F ε ( v ε ∗ φ ε , Q n ) lim Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Vector Case: the 1D Profile is Not Optimal F flat ( u 0 , Q ) = lim ε → 0 F ε ( u 0 ∗ φ ε , Q ) = γ 11 ( e 1 )+ γ 22 ( e 1 ) (1,1) u 0 (0,0) = Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Vector Case: the 1D Profile is Not Optimal F flat ( u 0 , Q ) = lim ε → 0 F ε ( u 0 ∗ φ ε , Q ) = γ 11 ( e 1 )+ γ 22 ( e 1 ) (1,1) u 0 (0,0) = Consider δ (1,0) (0,0) v δ v = = (0,0) (1,1) (1,1) δ → 0 v δ = u 0 lim Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
A Vector Case: the 1D Profile is Not Optimal F flat ( u 0 , Q ) = lim ε → 0 F ε ( u 0 ∗ φ ε , Q ) = γ 11 ( e 1 )+ γ 22 ( e 1 ) (1,1) u 0 (0,0) = Consider δ (1,0) (0,0) v δ v = = (0,0) (1,1) (1,1) δ → 0 v δ = u 0 lim ε → 0 F ε ( v δ ε ∗ φ ε , Q ) < F flat ( u 0 , Q ) lim Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Approximation: A First Step Model Problem: a One-Dimensional Scalar Isotropic Functional by Alberti-Bouchitt´ e-Seppecher Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Approximation: A First Step Model Problem: a One-Dimensional Scalar Isotropic Functional by Alberti-Bouchitt´ e-Seppecher � | u ( x ) − u ( y ) | 2 � 1 1 1 dx dy + 1 � � � 2 ( I ) I W ( u ) dx if u ∈ H | x − y | 2 | log ε | 2 I I ε F ε ( u ) = otherwise in L 1 ( I ) + ∞ with I = (0 , 1), u : I → R , W : R → [0 , + ∞ ), { W = 0 } = Z Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Approximation: A First Step Model Problem: a One-Dimensional Scalar Isotropic Functional by Alberti-Bouchitt´ e-Seppecher � | u ( x ) − u ( y ) | 2 � 1 1 1 dx dy + 1 � � � 2 ( I ) I W ( u ) dx if u ∈ H | x − y | 2 | log ε | 2 I I ε F ε ( u ) = otherwise in L 1 ( I ) + ∞ with I = (0 , 1), u : I → R , W : R → [0 , + ∞ ), { W = 0 } = Z F ε Γ( L 1 )-converges to Su | [ u ] | d H 0 � if u ∈ BV ( I , Z ) F ( u ) = otherwise in L 1 ( I ) . + ∞ Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Approximation: Finite Elements Method For every h > 0 we denote by I h the partition of I = (0 , 1) in intervals of width h and consider the Linear Finite Elements Space � � V h ( I ) = u : I → R : u ∈ C ( I ) , u | I ∈ P 1 ( I ) ∀ I ∈ I h . Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Approximation: Finite Elements Method For every h > 0 we denote by I h the partition of I = (0 , 1) in intervals of width h and consider the Linear Finite Elements Space � � V h ( I ) = u : I → R : u ∈ C ( I ) , u | I ∈ P 1 ( I ) ∀ I ∈ I h . Let { φ i } be a basis for V h . Every u ∈ V h ( I ) writes as N ( h ) � u ( x ) = u i φ i ( x ) u ← → U := { u 1 , ..., u N } , i =1 where N ( h ) is the (finite) dimension of V h ( I ). Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Approximation: Finite Elements Method For every h > 0 we denote by I h the partition of I = (0 , 1) in intervals of width h and consider the Linear Finite Elements Space � � V h ( I ) = u : I → R : u ∈ C ( I ) , u | I ∈ P 1 ( I ) ∀ I ∈ I h . Let { φ i } be a basis for V h . Every u ∈ V h ( I ) writes as N ( h ) � u ( x ) = u i φ i ( x ) u ← → U := { u 1 , ..., u N } , i =1 where N ( h ) is the (finite) dimension of V h ( I ). We define the bilinear form on V h ( I ): � �� � u ( x ) − u ( y ) v ( x ) − v ( y ) � � A ( u , v ) := dx dy . | x − y | 2 I I Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Approximation: Finite Elements Method It follows that | u ( x ) − u ( y ) | 2 � � dx dy = A ( u , u ) = U T A h U | x − y | 2 I I with A h = ( A ij h ) = ( A ( φ i , φ j )) Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Approximation: Finite Elements Method We set � � � W h ( U ) = π h W ( u ) dx I where π h : C ( I ) → V h ( I ) denotes the Lagrange interpolation operator. Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Approximation: Finite Elements Method We set � � � W h ( U ) = π h W ( u ) dx I where π h : C ( I ) → V h ( I ) denotes the Lagrange interpolation operator. We then define the following discrete approximation of the functional F ε : 1 � 1 2 U T A h U + 1 � F ε, h ( u ) = ε W h ( U ) | log ε | Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Approximation: Finite Elements Method We set � � � W h ( U ) = π h W ( u ) dx I where π h : C ( I ) → V h ( I ) denotes the Lagrange interpolation operator. We then define the following discrete approximation of the functional F ε : 1 � 1 2 U T A h U + 1 � F ε, h ( u ) = ε W h ( U ) | log ε | If h = h ( ε ) = o ( ε ) we expect that F ε, h Γ-converges to the same limit functional F : � Γ | [ u ] | d H 0 F ε, h ( u ) − → F ( u ) = Su Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Tests We look for Local Minimizers of F ε, h by means of the Gradient Descent Method Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Tests We look for Local Minimizers of F ε, h by means of the Gradient Descent Method ε = 10 − 2 h = 10 − 3 1 0.8 0.6 u 0 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Tests We look for Local Minimizers of F ε, h by means of the Gradient Descent Method ε = 10 − 2 h = 10 − 3 1 0.8 0.6 u ε 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F ε, h ( u ε ) = 1 . 0173 Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Tests We look for Local Minimizers of F ε, h by means of the Gradient Descent Method ε = 10 − 2 h = 10 − 3 1 0.8 0.6 u 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F ε, h ( u ε ) = 1 . 0173 ∼ 1 = F ( u ) Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Tests We look for Local Minimizers of F ε, h by means of the Gradient Descent Method ε = 10 − 2 h = 10 − 3 2 1.8 1.6 1.4 1.2 u 0 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Tests We look for Local Minimizers of F ε, h by means of the Gradient Descent Method ε = 10 − 2 h = 10 − 3 2 1.8 1.6 1.4 1.2 u ε 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F ε, h ( u ε ) = 1 . 9593 Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Tests We look for Local Minimizers of F ε, h by means of the Gradient Descent Method ε = 10 − 2 h = 10 − 3 2 1.8 1.6 1.4 1.2 u 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F ε, h ( u ε ) = 1 . 9593 ∼ 2 = F ( u ) Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Tests We look for Local Minimizers of F ε, h by means of the Gradient Descent Method ε = 10 − 2 h = 10 − 3 3 2.5 2 u 0 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Tests We look for Local Minimizers of F ε, h by means of the Gradient Descent Method ε = 10 − 2 h = 10 − 3 3 2.5 2 u ε 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F ε, h ( u ε ) = 2 . 9919 Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Tests We look for Local Minimizers of F ε, h by means of the Gradient Descent Method ε = 10 − 2 h = 10 − 3 3 2.5 2 u 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F ε, h ( u ε ) = 2 . 9919 ∼ 3 = F ( u ) Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
Numerical Tests We look for Local Minimizers of F ε, h by means of the Gradient Descent Method ε = 10 − 2 h = 10 − 3 3 2.5 2 u 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 The logarithmic rescaling slows down the convergence of the energy F ε, h Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
To Do ... Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure
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